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The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations... 1IntroductionLet Ω⊂RN\Omega \subset {R}^{N}be a bounded domain with smooth boundary (i.e., the derivative of the function at the boundary exists and is continuous). In this paper, we study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω\Omega : (1.1)utt+a(x)(−Δ)mut+b(x)M(‖∇mu‖)(−Δ)mu+g(x,u)=f(x,t)+h(x)∂w∂t,u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ,u(x,τ)=uτ(x),ut(x,τ)=u1τ(x),x∈Ω,\left\{\begin{array}{l}{u}_{tt}+a\left(x){\left(-\Delta )}^{m}{u}_{t}+b\left(x)M\left(\Vert {\nabla }^{m}u\Vert ){\left(-\Delta )}^{m}u+g\left(x,u)=f\left(x,t)+h\left(x)\frac{\partial w}{\partial t},\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{1em}i=1,2,\ldots ,m-1,\hspace{1em}x\in \Gamma ,\hspace{1em}t\ge \tau ,\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}{u}_{t}\left(x,\tau )={u}_{1\tau }\left(x),\hspace{1em}x\in \Omega ,\end{array}\right.where Γ\Gamma is the smooth boundary of Ω\Omega , vvis the outer normal vector on the boundary Γ\Gamma , m>1m\gt 1, a(x)andb(x)a\left(x)\hspace{0.25em}\text{and}\hspace{0.25em}b\left(x)are variable coefficient functions, f(x,t)∈Lloc2(R,Vk(Ω))f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega ))is a time-dependent external force term, wwis a one-dimensional bilateral standard Wiener process, h(x)∂w∂th\left(x)\frac{\partial w}{\partial t}describes white noise, and g(x,u)g\left(x,u)is a nonlinear function that satisfies certain growth conditions and dissipation conditions.The Kirchhoff model was proposed in 1883 to describe the motion of elastic cross-section. Compared with classical wave equations, the Kirchhoff model can describe the motion of elastic rod more accurately. There has been a lot of in-depth research on the Kirchhoff equations. [1,2,3, 4,5,6] studied the long-term dynamics of the autonomous low-order Kirchhoff equation; [7,8,9, 10,11] studied the existence of global solutions and the blow-up of solutions of the higher-order Kirchhoff equations.Stochastic wave equations are a very important class of stochastic partial differential equations, which are widely used in many fields such as fluid mechanics, physics, electricity, etc. The random attractor is an important tool for studying the long-term asymptotic behavior of stochastic dynamical systems. Using it to characterize the long-term behavior of random dynamical systems has laid a solid foundation for the study of random dynamical systems. After more than 30 years of development, random dynamical systems have also been extensively studied. Many scholars have conducted in-depth studies on the dynamical behavior of random wave equations in unbounded domains [12,13,14, 15,16,17] and bounded domains [18,19,20]. For new trends in functional analysis and random attractors, see also [21,22, 23,24].Regarding the variable coefficients in the equation, it represents the wave velocity at the space coordinate xx, which will appear in the wave phenomena in mathematical physics, marine acoustics, and other fields. It is of great practical significance to study the mathematical and physical equations with variable coefficients. In [25], they studied the global well-posedness and asymptotic behavior of solutions of Kirchhoff-type equations with variable coefficients and weak damping in unbounded domains. More relevant results can also be found in [26,27,28, 29,30,31, 32,33].In recent years, Lin and Chen [34], Lin and Jin [35] have performed a detailed study on the long-term dynamical behavior of higher-order wave equations and proposed the concept of the family of attractors. Combined with the current research results, there are no relevant research results on the long-time dynamics of the nonautonomous stochastic higher-order Kirchhoff equation, and the asymptotic behavior of the higher-order Kirchhoff equation with variable coefficients has not been studied. By studying the nonautonomous stochastic higher-order Kirchhoff model with variable coefficients, the relevant results of the Kirchhoff model can be generalized, and the theoretical achievements of the Kirchhoff model can be enriched, which lays a theoretical foundation for later application. Therefore, this article will specifically study the family of random attractors of nonautonomous random higher-order Kirchhoff equation with variable coefficients. In the research process, the reasonable assumption and Leibniz formula are used to overcome how to define the Lp{L}^{p}-weighted space and the difficulty of estimating the absorption sets and asymptotic compactness caused by the variable coefficients.Section 2 of this article introduces related theories, related definitions, and theories of stochastic dynamical systems; Section 3 presents the family of the continuous cocycle of the problem; In Section 4, the uniform estimation of the solution of problem (1.1) is obtained, and the asymptotic compactness of Φk{\Phi }_{k}is obtained through the decomposition method; in Section 5, we get the family of Dk{{\mathcal{D}}}_{k}-random attractors of Φk{\Phi }_{k}in Xk{X}_{k}.2Preparatory knowledgeIn this section, we mainly give the related theories of nonautonomous stochastic dynamical systems and random attractor (the family of random attractors).First, the relevant notation needed in this paper is introduced: Define the inner product and norm on H=L2(Ω)H={L}^{2}\left(\Omega )as (⋅,⋅)\left(\cdot ,\cdot )and (‖⋅‖)\left(\Vert \cdot \Vert ), Lp=Lp(Ω),‖⋅‖p=‖⋅‖Lp{L}^{p}={L}^{p}\left(\Omega ),\Vert \cdot {\Vert }_{p}=\Vert \cdot {\Vert }_{{L}^{p}}, where p≥1p\ge 1. Set variable coefficient a(x),b(x)=b0a(x),b0a\left(x),b\left(x)={b}_{0}a\left(x),{b}_{0}as a positive constant, satisfying a∈C0∞(Ω),a(x)≥a00>0,∂ia∂iv∣Γ=0,a0=‖a(x)‖∞,a(x)−1=μ(x),x∈Ωa\in {C}_{0}^{\infty }\left(\Omega ),a\left(x)\ge {a}_{00}\gt 0,\frac{{\partial }^{i}a}{{\partial }^{i}v}{| }_{\Gamma }=0,{a}_{0}=\Vert a\left(x){\Vert }_{\infty },a{\left(x)}^{-1}=\mu \left(x),x\in \Omega , and μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ).By D1,2{D}^{1,2}, we define the closure of the C0∞(Ω){C}_{0}^{\infty }\left(\Omega )functions with respect to the “energy norm” ‖u‖D1,2=∫Ω∣∇u∣2dx\Vert u{\Vert }_{{D}^{1,2}}={\int }_{\Omega }| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x. It is well known that D1,2≡D1,2(Ω)={u∈L2N/(N−2)(Ω)∣∇u∈(L2(Ω))N},{D}^{1,2}\equiv {D}^{1,2}\left(\Omega )=\left\{u\in {L}^{2N\text{/}\left(N-2)}\left(\Omega )| \nabla u\in {\left({L}^{2}\left(\Omega ))}^{N}\right\},and for D1,2↪L2N/(N−2)(Ω){D}^{1,2}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{2N\text{/}\left(N-2)}\left(\Omega ), there exists β>0\beta \gt 0such that ‖u‖2N/(N−2)≤β‖u‖D1,2\Vert u{\Vert }_{2N\text{/}\left(N-2)}\le \beta \Vert u{\Vert }_{{D}^{1,2}}.Lemma 2.1[26] Suppose that μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ), then for all u∈C0∞(Ω)u\in {C}_{0}^{\infty }\left(\Omega ), there exists α>0\alpha \gt 0such thatα∫Ωμu2dx≤∫Ω∣∇u∣2dx,\alpha \mathop{\int }\limits_{\Omega }\mu {u}^{2}{\rm{d}}x\le \mathop{\int }\limits_{\Omega }| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x,where α=β−2‖μ‖N/2−1\alpha ={\beta }^{-2}\Vert \mu {\Vert }_{N\hspace{-0.08em}\text{/}\hspace{-0.08em}2}^{-1}.Let μ>0\mu \gt 0be the weight function, and the weighted space Lμp=Lμp(Ω){L}_{\mu }^{p}={L}_{\mu }^{p}\left(\Omega )with the following norm: ‖u‖Lμpp=∫Ωμ∣u∣pdx=‖μ1pu‖pp,\Vert u{\Vert }_{{L}_{\mu }^{p}}^{p}=\mathop{\int }\limits_{\Omega }\mu | u\hspace{-0.25em}{| }^{p}{\rm{d}}x=\Vert {\mu }^{\tfrac{1}{p}}u{\Vert }_{p}^{p},for 1≤p<+∞1\le p\lt +\infty . Clearly Lμ2=Lμ2(Ω){L}_{\mu }^{2}={L}_{\mu }^{2}\left(\Omega )is a separable Hilbert space the inner product and norm are respectively:(u,v)μ=∫Ωμuvdx=μ12u,μ12v,‖u‖Lμ2=‖μ12u‖.{\left(u,v)}_{\mu }=\mathop{\int }\limits_{\Omega }\mu uv{\rm{d}}x=\left({\mu }^{\tfrac{1}{2}}u,{\mu }^{\tfrac{1}{2}}v\right),\hspace{1em}\Vert u{\Vert }_{{L}_{\mu }^{2}}=\Vert {\mu }^{\tfrac{1}{2}}u\Vert .For p:1≤p<∞p:1\le p\lt \infty , the Banach space Lμp{L}_{\mu }^{p}is uniformly convex, reflexive space, and (Lμp)′=Lμp′{\left({L}_{\mu }^{p})}^{^{\prime} }={L}_{\mu }^{p^{\prime} }, where p′{p}^{^{\prime} }is the conjugate number of pp.Lemma 2.2[26] Suppose that μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ), then D1,2{D}^{1,2}is compactly embedded in Lμ2{L}_{\mu }^{2}. LetVm=H0m(Ω)=Hm(Ω)∩H01(Ω),Vm+k=H0m+k(Ω)=Hm+k(Ω)∩H01(Ω),k=0,1,…,m,{V}_{m}={H}_{0}^{m}\left(\Omega )={H}^{m}\left(\Omega )\cap {H}_{0}^{1}\left(\Omega ),\hspace{1em}{V}_{m+k}={H}_{0}^{m+k}\left(\Omega )={H}^{m+k}\left(\Omega )\cap {H}_{0}^{1}\left(\Omega ),\hspace{1em}k=0,1,\ldots ,m,and the corresponding inner product and norm are, respectively,(u,v)Vm+k=(∇m+ku,∇m+kv),‖u‖Vm+k=‖∇m+ku‖H.{\left(u,v)}_{{V}_{m+k}}=\left({\nabla }^{m+k}u,{\nabla }^{m+k}v),\hspace{1em}\Vert u{\Vert }_{{V}_{m+k}}=\Vert {\nabla }^{m+k}u{\Vert }_{H}.At the same time, a general form of Poincare inequality: λ1‖∇ru‖2≤‖∇r+1u‖2{\lambda }_{1}\Vert {\nabla }^{r}u{\Vert }^{2}\le \Vert {\nabla }^{r+1}u{\Vert }^{2}, where λ1{\lambda }_{1}is the first eigenvalue of −Δ-\Delta . In the text, Ci{C}_{i}is a positive constant, C(⋅)C\left(\cdot )represents a positive constant that depends on the parameters in parentheses, and Cmn{C}_{m}^{n}is the corresponding number of combinations.Assuming that (X,‖⋅‖X)\left(X,\Vert \cdot {\Vert }_{X})is a separable Hilbert space, and B(X)B\left(X)is the Borel σ\sigma -algebra of X(Ω1,ℱ,P)X\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P)is the metric probability space.Definition 2.3[12] Let θt:R×Ω1→Ω1{\theta }_{t}:R\times {\Omega }_{1}\to {\Omega }_{1}be a family of (B(X)×ℱ,ℱ)\left(B\left(X)\times {\mathcal{ {\mathcal F} }},{\mathcal{ {\mathcal F} }})-measurable mappings such that θ0(⋅){\theta }_{0}\left(\cdot )is the identity on Ω1∀t,s∈R,θt+s(⋅)=θt(⋅)∘θs(⋅),Pθt(⋅)=P{\Omega }_{1}\hspace{0.33em}\forall t,s\in R,{\theta }_{t+s}\left(\cdot )={\theta }_{t}\left(\cdot )\circ {\theta }_{s}\left(\cdot ),P{\theta }_{t}\left(\cdot )=P. A mapping Φ:R+×R×Ω1×X→X\Phi :{R}^{+}\times R\times {\Omega }_{1}\times X\to Xis called a continuous cocycle or continuous random dynamical system (RDS) on XXover RRand (Ω1,ℱ,P,(θt)t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left({\theta }_{t})}_{t\in R})if for all τ∈R,w∈Ω1,t,s∈R+\tau \in R,w\in {\Omega }_{1},t,s\in {R}^{+}the following conditions are satisfied: i:Φ(⋅,τ,⋅,⋅):R+×Ω1×X→X\Phi \left(\cdot ,\hspace{0.25em}\tau ,\cdot ,\cdot ):{R}^{+}\times {\Omega }_{1}\times X\to Xis a (B(R+)×ℱ×B(X),B(X))\left(B\left({R}^{+})\times {\mathcal{ {\mathcal F} }}\times B\left(X),B\left(X))-measurable mapping;ii:Φ(0,τ,w,⋅)\Phi \left(0,\tau ,w,\cdot )is the identity on XX;iii:Φ(t+s,τ,w,⋅)=Φ(t,τ+s,θsw,Φ(s,τ,w,⋅))\Phi \left(t+s,\tau ,w,\cdot )=\Phi \left(t,\tau +s,{\theta }_{s}w,\Phi \left(s,\tau ,w,\cdot ));iv:Φ(t,τ,w,⋅):X→X\Phi \left(t,\tau ,w,\cdot ):X\to Xis continuous.Let D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}be a family of subsets parameterized by (τ,w)∈R×Ω1\left(\tau ,w)\in R\times {\Omega }_{1}in XX.Definition 2.4[13] The family D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}satisfies: (1)for all (τ,w)∈R×Ω1D(τ,w)\left(\tau ,w)\in R\times {\Omega }_{1}\hspace{0.33em}D\left(\tau ,w)is a closed nonempty subset of XX;(2)for every fixed x∈Xx\in Xand any τ∈R\tau \in R, the mapping w∈Ω1→distX(x,B(τ,w))w\in {\Omega }_{1}\to {{\rm{dist}}}_{X}\left(x,B\left(\tau ,w))is (ℱ,B(R+))\left({\mathcal{ {\mathcal F} }},B\left({R}^{+}))measurable, then the family DDis measurable with to ℱ{\mathcal{ {\mathcal F} }}in Ω1{\Omega }_{1}.Definition 2.5[15] For all σ>0,w∈Ω1D={D(τ,w)⊆X:τ∈R,w∈Ω1}\sigma \gt 0,w\in {\Omega }_{1}\hspace{0.33em}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}satisfies: limt→−∞eσt‖D(τ+t,θtw)‖X=0,\mathop{\mathrm{lim}}\limits_{t\to -\infty }{e}^{\sigma t}\Vert D\left(\tau +t,{\theta }_{t}w){\Vert }_{X}=0,then D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}is called tempered.Let D=D(X){\mathcal{D}}={\mathcal{D}}\left(X)be the set of all random tempered sets in XX.Definition 2.6[12] A family K={K(τ,w)⊆X:τ∈R,w∈Ω1}∈DK=\left\{K\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}\in {\mathcal{D}}of nonempty subsets of XXis called a measurable D{\mathcal{D}}-pullback attracting(or absorbing) set for {Φ(t,τ,w)}t≥0,τ∈R,w∈Ω1{\left\{\Phi \left(t,\tau ,w)\right\}}_{t\ge 0,\tau \in R,w\in {\Omega }_{1}}if (1)KKis measurable with respect to the PPcompletion of ℱ{\mathcal{ {\mathcal F} }}in Ω1{\Omega }_{1};(2)for all τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}and for every D∈DD\in {\mathcal{D}}, there exists T(D,τ,w)>0T\left(D,\tau ,w)\gt 0such that Φ(t,τ−t,θ−tw,D(τ−t,θ−tw))⊆K(τ,w),∀t≥T(D,τ,w).\Phi \left(t,\tau -t,{\theta }_{-t}w,D\left(\tau -t,{\theta }_{-t}w))\subseteq K\left(\tau ,w),\hspace{1em}\forall t\ge T\left(D,\tau ,w).Definition 2.7[15] Φ\Phi is said to be asymptotically compact in XXif for τ∈R,w∈Ω1,D={D(τ,w)⊆X:τ∈R,w∈Ω1}∈D,xn∈B(τ−tn,θ−tnw){Φ(tn,τ−tn,θ−tnw,xn)}n=1∞\tau \in R,w\in {\Omega }_{1},D=\{D\left(\tau ,w)\hspace{0.25em}\subseteq X:\tau \in R,w\in {\Omega }_{1}\}\in {\mathcal{D}},{x}_{n}\in B\left(\tau -{t}_{n},{\theta }_{-{t}_{n}}w)\hspace{0.33em}{\left\{\Phi \left({t}_{n},\tau -{t}_{n},{\theta }_{-{t}_{n}}w,{x}_{n})\right\}}_{n=1}^{\infty }has a convergent subsequence in XXwhenever tn→∞{t}_{n}\to \infty .Definition 2.8[13] A family A={A(τ,w)⊆X:τ∈R,w∈Ω1}∈DA=\left\{A\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}\in {\mathcal{D}}is called a D{\mathcal{D}}-pullback random attractor for {Φ(t,τ,w)}t≥0,τ∈R,w∈Ω1{\left\{\Phi \left(t,\tau ,w)\right\}}_{t\ge 0,\tau \in R,w\in {\Omega }_{1}}if (1)A(τ,w)A\left(\tau ,w)is measurable in Ω1{\Omega }_{1}with respect to ℱ{\mathcal{ {\mathcal F} }}and compact in XXfor ∀τ∈R,w∈Ω1\forall \hspace{-0.25em}\tau \in R,w\in {\Omega }_{1},(2)AAis invariant, i.e., for ∀τ∈R\forall \hspace{-0.25em}\tau \in Rand w∈Ω1,∀t≥0w\in {\Omega }_{1},\forall t\ge 0, Φ(t,τ,w,A(τ,w))=A(t+τ,θtw);\Phi \left(t,\tau ,w,A\left(\tau ,w))=A\left(t+\tau ,{\theta }_{t}w);(3)AAattracts every member of D{\mathcal{D}}, i.e., for every D∈D,τ∈RD\in {\mathcal{D}},\tau \in Rand for every w∈Ω1w\in {\Omega }_{1}, limt→+∞distX(Φ(t,τ−t,θ−tw,B(τ−t,θ−tw)),A(τ,w))=0,\mathop{\mathrm{lim}}\limits_{t\to +\infty }{{\rm{dist}}}_{X}\left(\Phi \left(t,\tau -t,{\theta }_{-t}w,B\left(\tau -t,{\theta }_{-t}w)),A\left(\tau ,w))=0,where distX(P,Q){{\rm{dist}}}_{X}\left(P,Q)denotes the Hausdorff semi-distance between two subsets PPand QQof XX.If we change D=D(X){\mathcal{D}}={\mathcal{D}}\left(X)to Dk=Dk(Xk){{\mathcal{D}}}_{k}={{\mathcal{D}}}_{k}\left({X}_{k}), where k=0,1,…,mk=0,1,\ldots ,m, then AAin Definition 2.8 can be a family of random attractors {Ak}\left\{{A}_{k}\right\}.Lemma 2.9[12] Let D{\mathcal{D}}be a neighborhood-closed collection of (τ,w)\left(\tau ,w)-parametrized families of nonempty subsets of XXand Φ\Phi be a continuous cocycle on XXover RRand (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R}), then Φ\Phi has a pullback D{\mathcal{D}}-attract AAif and only if Φ\Phi is pullback D{\mathcal{D}}asymptotically compact in XXand Φ\Phi has a closed, ℱ{\mathcal{ {\mathcal F} }}-measurable pullback D{\mathcal{D}}-absorbing set KKin D{\mathcal{D}}and the unique pullback D{\mathcal{D}}-attractor A={A(τ,w)}A=\left\{A\left(\tau ,w)\right\}is given byA(τ,w)=⋂τ≥0⋃t≥τΦ(t,τ−t,θ−tw,K(τ−t,θ−tw))¯,τ∈R,w∈Ω1.A\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }\Phi \left(t,\tau -t,{\theta }_{-t}w,K\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.Similarly, Lemma 2.9 can be extended to Lemma 2.10 of the family of pullback attractors.Lemma 2.10Let Dk{{\mathcal{D}}}_{k}be neighborhood-closed collections of (τ,w)\left(\tau ,w)-parametrized families of nonempty subsets of Xk,k=1,2,…,m{X}_{k},k=1,2,\ldots ,m, and Φk{\Phi }_{k}be the family of continuous cocycles on Xk,k=1,2,…,m{X}_{k},k=1,2,\ldots ,mover RRand (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R}), then Φk{\Phi }_{k}has the family of pullback Dk{{\mathcal{D}}}_{k}-attracts {Ak}\left\{{A}_{k}\right\}if and only if Φk{\Phi }_{k}is pullback Dk{{\mathcal{D}}}_{k}-asymptotically compact in Xk{X}_{k}and Dk{{\mathcal{D}}}_{k}has closed, ℱ{\mathcal{ {\mathcal F} }}-measurable pullback Dk{{\mathcal{D}}}_{k}-absorbing sets Kk{K}_{k}in Dk{{\mathcal{D}}}_{k}and the unique pullback Dk{{\mathcal{D}}}_{k}-attractor Ak={Ak(τ,w)}{A}_{k}=\left\{{A}_{k}\left(\tau ,w)\right\}is given byAk(τ,w)=⋂τ≥0⋃t≥τΦk(t,τ−t,θ−tw,Kk(τ−t,θ−tw))¯,τ∈R,w∈Ω1.{A}_{k}\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }{\Phi }_{k}\left(t,\tau -t,{\theta }_{-t}w,{K}_{k}\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.3The family of cocycles of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficientsLet (Ω1,ℱ,P)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P)be a probability space, where Ω1={w∈C(R,R),w(0)=0}.{\Omega }_{1}=\left\{w\in C\left(R,R),w\left(0)=0\right\}.wwis a two-sided real-valued Winner processes on the probability space (Ω1,ℱ,P)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P). Define θtw(⋅)=w(⋅+t)−w(t),w∈Ω1,t∈R{\theta }_{t}w\left(\cdot )=w\left(\cdot +t)-w\left(t),w\in {\Omega }_{1},t\in R, thus, (Ω1,ℱ,P,(θt)t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left({\theta }_{t})}_{t\in R})is an ergodic metric dynamical system.For a small positive number ε\varepsilon , let zzbe a new variable given by z=ut+εuz={u}_{t}+\varepsilon uand then, system (1.1) becomes (3.1)∂u∂t+εu=z;∂z∂t=εz−a(x)(−Δ)mz+εa(x)(−Δ)mu−ε2u−b(x)M(‖∇mu‖2)(−Δ)mu−g(x,u)+f(x,t)+h(x)dwdt;u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ;u(x,τ)=uτ(x),z(x,τ)=zτ(x)=u1τ(x)+uτ(x),x∈Ω,\left\{\begin{array}{l}\frac{\partial u}{\partial t}+\varepsilon u=z;\\ \frac{\partial z}{\partial t}=\varepsilon z-a\left(x){\left(-\Delta )}^{m}z+\varepsilon a\left(x){\left(-\Delta )}^{m}u-{\varepsilon }^{2}u-b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u-g\left(x,u)+f\left(x,t)+h\left(x)\frac{{\rm{d}}w}{{\rm{d}}t};\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{0.33em}i=1,2,\ldots ,m-1,\hspace{0.33em}x\in \Gamma ,\hspace{0.33em}t\ge \tau ;\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}z\left(x,\tau )={z}_{\tau }\left(x)={u}_{1\tau }\left(x)+{u}_{\tau }\left(x),\hspace{0.33em}x\in \Omega ,\end{array}\right.where (M)M∈C1(R+),M′≥0\left(M)M\in {C}^{1}\left({R}^{+}),M^{\prime} \ge 0, and M(s)≤M0⋅(1+sq),0<q<1/2,M0=M(0)M\left(s)\le {M}_{0}\cdot \left(1+{s}^{q}),0\lt q\lt 1\hspace{-0.08em}\text{/}\hspace{-0.08em}2,{M}_{0}=M\left(0)is a positive constant ∀s∈R+\forall \hspace{-0.25em}s\in {R}^{+}, h(x)∈Vm+k(Ω),x∈Ω,t≥τ,τ∈R,k=0,1,…,m,f(x,t)∈Lloc2(R,Vk(Ω))h\left(x)\in {V}_{m+k}\left(\Omega ),x\in \Omega ,t\ge \tau ,\tau \in R,k=0,1,\ldots ,m,f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega )). In order to get the conclusion of this article, suppose that the nonlinear term g(x,u)g\left(x,u)satisfies the following conditions: for ∀u∈R,x∈Ω\forall \hspace{-0.25em}u\in R,x\in \Omega , there are positive constants c1,c2,c3,c4,c5>0{c}_{1},{c}_{2},{c}_{3},{c}_{4},{c}_{5}\gt 0, satisfying (3.2)∣g(x,u)∣≤c1∣p∣p+ϕ1(x),ϕ1∈Lμ2(Ω),\hspace{-17.6em}| g\left(x,u)| \le {c}_{1}| p\hspace{-0.25em}{| }^{p}+{\phi }_{1}\left(x),\hspace{0.33em}{\phi }_{1}\in {L}_{\mu }^{2}\left(\Omega ),(3.3)ug(x,u)−c2G(x,u)≥ϕ2(x),ϕ2∈Lμ1(Ω),ug\left(x,u)-{c}_{2}G\left(x,u)\ge {\phi }_{2}\left(x),\hspace{0.33em}{\phi }_{2}\in {L}_{\mu }^{1}\left(\Omega ),(3.4)G(x,u)≥c3∣u∣p+1−ϕ3(x),ϕ3∈Lμ1(Ω),\hspace{-17.6em}G\left(x,u)\ge {c}_{3}| u\hspace{-0.25em}{| }^{p+1}-{\phi }_{3}\left(x),\hspace{0.33em}{\phi }_{3}\in {L}_{\mu }^{1}\left(\Omega ),(3.5)∣gu(x,u)∣≤c4∣u∣p−1+ϕ4(x),ϕ4∈Vm(Ω),\hspace{-17.6em}| {g}_{u}\left(x,u)| \le {c}_{4}| u\hspace{-0.25em}{| }^{p-1}+{\phi }_{4}\left(x),\hspace{0.33em}{\phi }_{4}\in {V}_{m}\left(\Omega ),(3.6)∣∇xkg(x,u)∣≤c5∣u∣p+ϕ5(x),ϕ5∈Vk(Ω),\hspace{-17.6em}| {\nabla }_{x}^{k}g\left(x,u)| \le {c}_{5}| u\hspace{-0.25em}{| }^{p}+{\phi }_{5}\left(x),\hspace{0.33em}{\phi }_{5}\in {V}_{k}\left(\Omega ),where 1≤p<+∞1\le p\lt +\infty , for N=1,2;N=1,2;1≤p<NN−21\le p\lt \frac{N}{N-2}N=3,4N=3,4; and G(x,u)=∫0ug(x,s)dsG\left(x,u)={\int }_{0}^{u}g\left(x,s){\rm{d}}s. From equations (3.2) and (3.3), we can get (3.7)G(x,u)≤c6(∣u∣2+∣u∣p+1+ϕ12+ϕ2).G\left(x,u)\le {c}_{6}\left(| u\hspace{-0.25em}{| }^{2}+| u\hspace{-0.25em}{| }^{p+1}+{\phi }_{1}^{2}+{\phi }_{2}).To show that problem (3.1) generates a random dynamical system, we let v(t,τ,w)=z(t,τ,w)−hw(t)v\left(t,\tau ,w)=z\left(t,\tau ,w)-hw\left(t), and then, (3.1) can be rewritten as the equivalent system with random coefficients but without white noise: (3.8)∂u∂t−v+εu=hw(t);∂v∂t=εv−a(x)(−Δ)mv+εa(x)(−Δ)mu−ε2u−b(x)M(‖∇mu‖2)(−Δ)mu−g(x,u)+f(x,t)+εh(x)w(t)−a(x)(−Δ)mh(x)w(t);u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ;u(x,τ)=uτ(x),v(x,τ)=vτ(x)=zτ(x)−hw(τ),x∈Ω.\left\{\begin{array}{l}\frac{\partial u}{\partial t}-v+\varepsilon u=hw\left(t);\\ \frac{\partial v}{\partial t}=\varepsilon v-a\left(x){\left(-\Delta )}^{m}v+\varepsilon a\left(x){\left(-\Delta )}^{m}u-{\varepsilon }^{2}u-b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u\\ -g\left(x,u)+f\left(x,t)+\varepsilon h\left(x)w\left(t)-a\left(x){\left(-\Delta )}^{m}h\left(x)w\left(t);\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{0.33em}i=1,2,\ldots ,m-1,\hspace{0.33em}x\in \Gamma ,\hspace{0.33em}t\ge \tau ;\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}v\left(x,\tau )={v}_{\tau }\left(x)={z}_{\tau }\left(x)-hw\left(\tau ),\hspace{0.33em}x\in \Omega .\end{array}\right.Let Xk=Vm+k×Vk,k=0,1,…,m{X}_{k}={V}_{m+k}\times {V}_{k},k=0,1,\ldots ,m, when k=0V0=Lμ2k=0\hspace{0.33em}{V}_{0}={L}_{\mu }^{2}, endowed with the usual norm ‖(u,v)‖Xk2=‖u‖Vm+k2+‖v‖Vk2\Vert \left(u,v){\Vert }_{{X}_{k}}^{2}=\Vert u{\Vert }_{{V}_{m+k}}^{2}+\Vert v{\Vert }_{{V}_{k}}^{2}. By the standard Galerkin method: If the assumptions (M)h(x)∈Vm+k(Ω),x∈Ω,t≥τ,τ∈R,f(x,t)∈Lloc2(R,Vk(Ω))\left(M)\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ),x\in \Omega ,t\ge \tau ,\tau \in R,f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega ))conditions (3.2)–(3.6) hold the problem (3.8) is well posed in Xk=Vm+k×Vk{X}_{k}={V}_{m+k}\times {V}_{k}, i.e., for all τ∈R\tau \in Rand P−a.e.w∈Ω1,(uτ,vτ)∈XkP-a.e.w\in {\Omega }_{1},\left({u}_{\tau },{v}_{\tau })\in {X}_{k}, the problem (3.8) has a unique global solution (u(t,τ,w,uτ),(u\left(t,\tau ,w,{u}_{\tau }),v(t,τ,w,vτ))∈C([τ,∞),Xk)v\left(t,\tau ,w,{v}_{\tau }))\in C\left({[}\tau ,\infty ),{X}_{k})and (u(τ,τ,w,uτ),v(τ,τ,w,vτ))=(uτ,vτ)\left(u\left(\tau ,\tau ,w,{u}_{\tau }),v\left(\tau ,\tau ,w,{v}_{\tau }))=\left({u}_{\tau },{v}_{\tau }). Moreover, for t≥τ,(u(t,τ,w,uτ),t\ge \tau ,(u\left(t,\tau ,w,{u}_{\tau }),v(t,τ,w,vτ))v\left(t,\tau ,w,{v}_{\tau }))is (ℱ,B(Xk))\left({\mathcal{ {\mathcal F} }},B\left({X}_{k}))measurable in wwand continuous in (uτ,vτ)\left({u}_{\tau },{v}_{\tau })with respect to the Xk{X}_{k}norm. Thus, the solution mapping can be used to define a family of continuous cocycles for (3.8). Let Φk:R+×R×Ω1×{\Phi }_{k}:{R}^{+}\times R\times {\Omega }_{1}\times Xk→Xk{X}_{k}\to {X}_{k}be mappings given by (3.9)Φk(t,τ,w,(uτ,vτ))=(u(t+τ,τ,θ−τw,uτ),v(t+τ,τ,θ−τw,vτ)),{\Phi }_{k}\left(t,\tau ,w,\left({u}_{\tau },{v}_{\tau }))=\left(u\left(t+\tau ,\tau ,{\theta }_{-\tau }w,{u}_{\tau }),v\left(t+\tau ,\tau ,{\theta }_{-\tau }w,{v}_{\tau })),where (t,τ,w,(uτ,vτ))∈R+×R×Ω1×Xk\left(t,\tau ,w,\left({u}_{\tau },{v}_{\tau }))\in {R}^{+}\times R\times {\Omega }_{1}\times {X}_{k}, then Φk{\Phi }_{k}is a family of continuous cocycles over (R,τ+t)\left(R,\tau +t)and (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R})on Xk{X}_{k}. For P−a.e.w∈Ω1P-a.e.w\in {\Omega }_{1}and t,s≥0,τ∈R:t,s\ge 0,\hspace{0.33em}\tau \in R:(3.10)Φk(t+s,τ,w,(uτ,vτ))=Φk(t,s+τ,w,Φk(s,τ,w,(uτ,vτ))).{\Phi }_{k}\left(t+s,\tau ,w,\left({u}_{\tau },{v}_{\tau }))={\Phi }_{k}\left(t,s+\tau ,w,{\Phi }_{k}\left(s,\tau ,w,\left({u}_{\tau },{v}_{\tau }))).For any bounded nonempty subset Bk{B}_{k}of Xk{X}_{k}denote by ‖Bk‖=supΦk∈R‖Φ‖Xk\Vert {B}_{k}\Vert ={\sup }_{{\Phi }_{k}\in R}\Vert \Phi {\Vert }_{{X}_{k}}. Let Dk={Dk(τ,w):τ∈R,w∈Ω1}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}be a family of bounded nonempty subsets of Xk{X}_{k}, and for all τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}, (3.11)lims→−∞eσs‖Dk(τ+s,θsw)‖Xk2=0.\mathop{\mathrm{lim}}\limits_{s\to -\infty }{e}^{\sigma s}\Vert {D}_{k}\left(\tau +s,{\theta }_{s}w){\Vert }_{{X}_{k}}^{2}=0.Remember that Dk{{\mathcal{D}}}_{k}is the set of the aforementioned subset family Dk{D}_{k}, that is, Dk={Dk={Dk(τ,w):{{\mathcal{D}}}_{k}=\{{D}_{k}=\{{D}_{k}\left(\tau ,w)\hspace{0.25em}:τ∈R,w∈Ω1}:Dksatisfies (3.11)}\tau \in R,w\in {\Omega }_{1}\}:{D}_{k}\hspace{0.33em}\hspace{0.1em}\text{satisfies (3.11)}\hspace{0.1em}\}.4Uniform estimates of solutionsTo prove the existence of the family of random attractors, we conduct uniform estimates on the solutions of the problem (3.8) defined on Ω\Omega , for the purposes of showing the existence of a family of Dk{{\mathcal{D}}}_{k}pullback absorbing sets and the pullback Dk{{\mathcal{D}}}_{k}asymptotic compactness of the random dynamical system. Let ε>0\varepsilon \gt 0be small enough and satisfy αλ1m−1−3ε>0,2a00λ1m−(a00λ1m+12)ε>0,M0−52ε>0,b0M08−ε>0\alpha {\lambda }_{1}^{m-1}-3\varepsilon \gt 0,2{a}_{00}{\lambda }_{1}^{m}-\left({a}_{00}{\lambda }_{1}^{m}+12)\varepsilon \gt 0,{M}_{0}-\frac{5}{2}\varepsilon \gt 0,\frac{{b}_{0}{M}_{0}}{8}-\varepsilon \gt 0, (4.1)σ=12minαλ1m−1−3ε,ε2,εc22,σ1=12min{2a00λ1m−(a00λ1m+12)ε,ε}.\sigma =\frac{1}{2}{\rm{\min }}\left\{\alpha {\lambda }_{1}^{m-1}-3\varepsilon ,\frac{\varepsilon }{2},\frac{\varepsilon {c}_{2}}{2}\right\},\hspace{1em}{\sigma }_{1}=\frac{1}{2}\min \left\{2{a}_{00}{\lambda }_{1}^{m}-\left({a}_{00}{\lambda }_{1}^{m}+12)\varepsilon ,\varepsilon \right\}.To obtain uniform estimates of the solutions, f(x,t)f\left(x,t)needs to satisfy (F1)∫−∞teσs‖f(⋅,s)‖Vk2ds<∞\left({F}_{1}){\int }_{-\infty }^{t}{e}^{\sigma s}\Vert f\left(\cdot ,\hspace{0.33em}s){\Vert }_{{V}_{k}}^{2}{\rm{d}}s\lt \infty .Lemma 4.1Suppose MMsatisfies (M),h(x)∈Vm+k(Ω),k=0,1,…,m\left(M),\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ),k=0,1,\ldots ,m, (3.2)–(3.6) hold, f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1}), and Bk={Bk(τ,w):τ∈R,w∈Ω1}∈Dk{B}_{k}=\left\{{B}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}for P−a.e.w∈Ω1,τ∈RP-a.e.w\in {\Omega }_{1},\tau \in Rinitial value satisfies (uτ−t,vτ−t)∈Bk(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{k}(\tau -t,{\theta }_{-\tau }w), there exists Tk=Tk(τ,w,Bk)>0{T}_{k}={T}_{k}\left(\tau ,w,{B}_{k})\gt 0such that for all t≥Tkt\ge {T}_{k}, the solution (u(τ,τ,w,uτ−t),v(τ,τ,w,vτ−t))=(uτ−t,vτ−t)(u\left(\tau ,\tau ,w,{u}_{\tau -t}),v(\tau ,\tau ,w,{v}_{\tau -t}))=\left({u}_{\tau -t},{v}_{\tau -t})of problem (3.8) satisfies‖v(τ,τ−t,θ−τw,vτ−t)‖Vk2+‖u(τ,τ−t,θ−τw,vτ−t)‖Vm+k2≤r1k(τ,w),\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{k}}^{2}+\Vert u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w),where r1k(τ,w){r}_{1k}\left(\tau ,w)will be given in detail later.ProofTaking the inner product of (3.8) with vvin Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ), we find that (4.2)12ddt‖v‖Lμ22=ε‖v‖Lμ22−‖∇mv‖2+ε((−Δ)mu,v)−ε2(u,v)Lμ2−b0(M(‖∇mu‖2)(−Δ)mu,v)−(g(x,u),v)Lμ2+(f(x,t),v)Lμ2+εw(t)(h,v)Lμ2−w(t)((−Δ)mh,v),\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}& =& \varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}-\Vert {\nabla }^{m}v{\Vert }^{2}+\varepsilon \left({\left(-\Delta )}^{m}u,v)-{\varepsilon }^{2}{\left(u,v)}_{{L}_{\mu }^{2}}-{b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,v)\\ & & -{\left(g\left(x,u),v)}_{{L}_{\mu }^{2}}+{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+\varepsilon w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-w\left(t)\left({\left(-\Delta )}^{m}h,v),\end{array}for each term on the right-hand side of (4.2): (4.3)ε((−Δ)mu,v)=ε((−Δ)mu,ut+εu+hw(t))=ε2ddt‖∇mu‖2+ε2‖∇mu‖2−εw(t)((−Δ)mu,h),\varepsilon \left({\left(-\Delta )}^{m}u,v)=\varepsilon \left({\left(-\Delta )}^{m}u,{u}_{t}+\varepsilon u+hw\left(t))=\frac{\varepsilon }{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}-\varepsilon w\left(t)\left({\left(-\Delta )}^{m}u,h),(4.4)ε2(u,v)Lμ2=ε2(u,ut+εu−hw(t))Lμ2=ε22ddt‖v‖Lμ22+ε3‖v‖Lμ22−ε2w(t)(u,h)Lμ2,\hspace{-35.1em}{\varepsilon }^{2}{\left(u,v)}_{{L}_{\mu }^{2}}={\varepsilon }^{2}{\left(u,{u}_{t}+\varepsilon u-hw\left(t))}_{{L}_{\mu }^{2}}=\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{3}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}-{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}},(4.5)b0(M(‖∇mu‖2)(−Δ)mu,v)=b0(M(‖∇mu‖2)(−Δ)mu,ut+εu−hw(t))=b02ddt∫0‖∇mu‖2M(s)ds+εb0M(‖∇mu‖2)‖∇mu‖2−b0M(‖∇mu‖2)w(t)((−Δ)mu,h),\hspace{-37.95em}\begin{array}{l}{b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,v)={b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{u}_{t}+\varepsilon u-hw\left(t))\\ \hspace{1.0em}=\frac{{b}_{0}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s)\hspace{0.33em}{\rm{d}}s+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}u{\Vert }^{2}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}u,h),\end{array}(4.6)(g(x,u),v)Lμ2=(g(x,u),ut+εu−hw(t))Lμ2=ddt∫ΩμG(x,u)dx+ε(g(x,u),u)Lμ2−w(t)(g(x,u),h)Lμ2.{\left(g\left(x,u),v)}_{{L}_{\mu }^{2}}={\left(g\left(x,u),{u}_{t}+\varepsilon u-hw\left(t))}_{{L}_{\mu }^{2}}=\frac{{\rm{d}}}{{\rm{d}}t}\mathop{\int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}-w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}.Substitute (4.3)–(4.6) into (4.2) to obtain (4.7)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+2‖∇mv‖2−2ε‖v‖Lμ22+2εb0M(‖∇mu‖2)‖∇mu‖2−2ε2‖∇mu‖2+2ε3‖u‖Lμ22+2ε(g(x,u),u)Lμ2=2(f(x,t),v)Lμ2+(2b0M(‖∇mu‖2)−2ε)w(t)((−Δ)mu,h)−2ε2w(t)(u,h)Lμ2+2w(t)(g(x,u),h)Lμ2+2w(t)(h,v)Lμ2−2w(t)((−Δ)mh,v).\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+2\Vert {\nabla }^{m}v{\Vert }^{2}\\ \hspace{1.0em}-2\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}u{\Vert }^{2}-2{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+2{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=2{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+\left(2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-2\varepsilon )w\left(t)\left({\left(-\Delta )}^{m}u,h)-2{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+2w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}+2w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-2w\left(t)\left({\left(-\Delta )}^{m}h,v).\end{array}Using the Cauchy-Schwarz inequality, Young’s inequality and Holder’s inequality, we have (4.8)2ε2w(t)(u,h)Lμ2≤ε3‖u‖Lμ22+ε∣w(t)∣2‖h‖Lμ22,2{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}}\le {\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},(4.9)2b0M(‖∇mu‖2)w(t)((−Δ)mu,h)≤2b0M0(1+‖∇mu‖2q)∣w(t)∣‖∇mu‖‖∇mh‖≤2b0M0∣w(t)∣‖∇mu‖‖∇mh‖+2b0M0∣w(t)∣‖∇mu‖2q+1‖∇mh‖≤εb0M04‖∇mu‖2+4ε−1b0M0∣w(t)∣2‖∇mh‖2+εb0M04‖∇mu‖2+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q.\begin{array}{l}2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}u,h)\\ \hspace{1.0em}\le \hspace{0.33em}2{b}_{0}{M}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})| w\left(t)| \Vert {\nabla }^{m}u\Vert \Vert {\nabla }^{m}h\Vert \\ \hspace{1.0em}\le 2{b}_{0}{M}_{0}| w\left(t)| \Vert {\nabla }^{m}u\Vert \Vert {\nabla }^{m}h\Vert +2{b}_{0}{M}_{0}| w\left(t)| \Vert {\nabla }^{m}u{\Vert }^{2q+1}\Vert {\nabla }^{m}h\Vert \\ \hspace{1.0em}\le \hspace{0.25em}\frac{\varepsilon {b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m}u{\Vert }^{2}+4{\varepsilon }^{-1}{b}_{0}{M}_{0}| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\frac{\varepsilon {b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m}u{\Vert }^{2}\\ \hspace{1.85em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}.\end{array}By (3.2) and (3.4), we get (4.10)2w(t)(g(x,u),h)Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣∫Ωμ∣u∣p+1dxpp+1‖h‖Lμp+1≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣∫Ω(μG(x,u)+μϕ3)dxpp+1‖h‖Lμp+1≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫ΩμG(x,u)dx+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1,\begin{array}{l}2w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| {\left(\mathop{\displaystyle \int }\limits_{\Omega }\mu | u{| }^{p+1}{\rm{d}}x\right)}^{\tfrac{p}{p+1}}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| {\left(\mathop{\displaystyle \int }\limits_{\Omega }\left(\mu G\left(x,u)+\mu {\phi }_{3}){\rm{d}}x\right)}^{\tfrac{p}{p+1}}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x\hspace{.85em}+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1},\end{array}(4.11)2(f(x,t),v)Lμ2+2w(t)(h,v)Lμ2−2w(t)((−Δ)mh,v)≤2‖f‖Lμ2‖v‖Lμ2+2ε∣w(t)∣‖h‖Lμ2‖v‖Lμ2+2∣w(t)∣‖∇mh‖‖∇mv‖≤2α−12λ1−m−12‖f‖Lμ2‖∇mv‖+ε‖v‖Lμ22+ε∣w(t)∣2‖h‖Lμ22+12‖∇mv‖2+2∣w(t)∣2‖∇mh‖2≤‖∇mv‖2+ε‖v‖Lμ22+2α−1λ11−m‖f‖Lμ22+2∣w(t)∣2‖∇mh‖2+ε∣w(t)∣2‖h‖Lμ22,\hspace{-35.5em}\begin{array}{l}2{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+2w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-2w\left(t)\left({\left(-\Delta )}^{m}h,v)\\ \hspace{1.0em}\le \hspace{-0.25em}2\Vert f{\Vert }_{{L}_{\mu }^{2}}\Vert v{\Vert }_{{L}_{\mu }^{2}}+2\varepsilon | w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}\Vert v{\Vert }_{{L}_{\mu }^{2}}+2| w\left(t)| \Vert {\nabla }^{m}h\Vert \Vert {\nabla }^{m}v\Vert \\ \hspace{1.0em}\le \hspace{-0.25em}2{\alpha }^{-\tfrac{1}{2}}{\lambda }_{1}^{-\frac{m-1}{2}}\Vert f{\Vert }_{{L}_{\mu }^{2}}\Vert {\nabla }^{m}v\Vert +\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+\frac{1}{2}\Vert {\nabla }^{m}v{\Vert }^{2}+2| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}\\ \hspace{1.0em}\le \Vert {\nabla }^{m}v{\Vert }^{2}+\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+2| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},\end{array}(4.12)2εw(t)((−Δ)mu,h)≤2ε∣w(t)∣‖∇mh‖‖∇mu‖≤ε2‖∇mu‖2+∣w(t)∣2‖∇mh‖2.\hspace{-35.5em}2\varepsilon w\left(t)\left({\left(-\Delta )}^{m}u,h)\le 2\varepsilon | w\left(t)| \Vert {\nabla }^{m}h\Vert \Vert {\nabla }^{m}u\Vert \le {\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}.Substitute (4.8)–(4.12) into (4.7) to obtain (4.13)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+‖∇mv‖2−3ε‖v‖Lμ22+2εM(‖∇mu‖2)−εM02b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+2ε(g(x,u),u)Lμ2\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+\Vert {\nabla }^{m}v{\Vert }^{2}\\ \hspace{1.0em}-3\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\frac{\varepsilon {M}_{0}}{2}\right){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}-3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\end{array}≤2α−1λ11−m‖f‖Lμ22+(3+4ε−1M0b0)∣w(t)∣2‖∇mh‖2+2ε∣w(t)∣2‖h‖Lμ22+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q+2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫ΩμG(x,u)dx+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1.\begin{array}{l}\hspace{1.0em}\le \hspace{-0.25em}2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}+2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\\ \hspace{2.0em}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}.\end{array}By condition (3.3), we have (4.14)2ε(g(x,u),u)Lμ2≥2εc2∫ΩμG(x,u)dx+∫Ωμϕ2(x)dx.2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\ge 2\varepsilon \left({c}_{2}\mathop{\int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\mathop{\int }\limits_{\Omega }\mu {\phi }_{2}\left(x){\rm{d}}x\right).Substitute (4.14) into (4.12) to obtain (4.15)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+‖∇mv‖2−3ε‖v‖Lμ22+2εM(‖∇mu‖2)−εM02b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+εc2∫ΩμG(x,u)dx+2ε∫Ωμϕ2(x)dx≤2α−1λ11−m‖f‖Lμ22+(3+4ε−1M0b0)∣w(t)∣2‖∇mh‖2+2ε∣w(t)∣2‖h‖Lμ22+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q+2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+\Vert {\nabla }^{m}v{\Vert }^{2}-3\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\frac{\varepsilon {M}_{0}}{2}\right){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}-3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+2\varepsilon \mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{2}\left(x){\rm{d}}x\\ \hspace{1.0em}\le 2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}+2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}.\end{array}According to (4.1), we get (4.16)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+σ‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx≤2α−1λ11−m‖f‖Lμ22+C01(1+∣w(t)∣2+∣w(t)∣21−2q+∣w(t)∣p+1),\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}+\sigma \left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s)\hspace{0.33em}{\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}\le 2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{01}\left(1+| w\left(t){| }^{2}+| w\left(t){| }^{\tfrac{2}{1-2q}}+| w\left(t){| }^{p+1}),\end{array}where C01=max(3+4ε−1M0b0)‖∇mh‖2+2ε‖h‖Lμ22+‖ϕ1‖Lμ22‖h‖Lμ22,εc2∫Ωμϕ3(x)dx,1−2q2ε8q+42q+12q−1b0M0‖∇mh‖21−2q,(2c1)p+1(εc2)−pp+1p+1p−p‖h‖Lμp+1p+1.\begin{array}{rcl}{C}_{01}& =& \max \left\{\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon \Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x,\right.\\ & & \left.\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}},{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}\right\}.\end{array}Using the Gronwall inequality to integrate (4.16) over [τ−t,τ]\left[\tau -t,\tau ]with t≥0t\ge 0and replacing wwby θ−τw{\theta }_{-\tau }w, we obtain (4.17)eστ‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx≤eσ(τ−t)‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx+2α−1λ11−m∫τ−tτeσξ‖f(⋅,ξ)‖Lμ22dξ+C01∫τ−tτeσξ(1+∣w(ξ)∣2+∣w(ξ)∣21−2q+∣w(ξ)∣p+1)dξ,\begin{array}{l}{e}^{\sigma \tau }\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}\le {e}^{\sigma \left(\tau -t)}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}\right.\\ \hspace{2.0em}\left.+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]+2{\alpha }^{-1}{\lambda }_{1}^{1-m}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \\ \hspace{1.0em}\hspace{1.0em}+{C}_{01}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\left(1+| w\left(\xi ){| }^{2}+| w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| w\left(\xi ){| }^{p+1}){\rm{d}}\xi ,\end{array}then (4.18)‖v(τ,τ−t,θ−τw,vτ−t)‖Lμ22+b0∫0‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2M(s)ds−ε‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2+ε2‖u(τ,τ−t,θ−τw,vτ−t)‖Lμ22+2∫ΩμG(x,u(τ,τ−t,θ−τw,vτ−t))dx≤e−σt‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx+2α−1λ11−me−στ∫τ−tτeσξ‖f(⋅,ξ)‖Lμ22dξ+C01e−στ∫τ−tτeσξ(1+∣θ−τw(ξ)∣2+∣θ−τw(ξ)∣21−2q+∣θ−τw(ξ)∣p+1)dξ.\begin{array}{l}\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}+{\varepsilon }^{2}\Vert u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t})){\rm{d}}x\\ \hspace{1.0em}\le {e}^{-\sigma t}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}\right.\\ \hspace{2.0em}\left.+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]+2{\alpha }^{-1}{\lambda }_{1}^{1-m}{e}^{-\sigma \tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \\ \hspace{2.0em}+{C}_{01}{e}^{-\sigma \tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\left(1+| {\theta }_{-\tau }w\left(\xi ){| }^{2}+| {\theta }_{-\tau }w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| {\theta }_{-\tau }w\left(\xi ){| }^{p+1}){\rm{d}}\xi .\end{array}By (3.7), we have (4.19)∫ΩμG(x,u(τ−t))dx≤c6(‖ϕ1‖Lμ22+‖ϕ2‖Lμ1+‖u‖Lμ22+‖u‖Lμp+1p+1)≤C02(1+‖u‖Lμ22+‖u‖Vmp+1).\mathop{\int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\le {c}_{6}\left(\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\phi }_{2}{\Vert }_{{L}_{\mu }^{1}}+\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert u{\Vert }_{{L}_{\mu }^{p+1}}^{p+1})\le {C}_{02}\left(1+\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert u{\Vert }_{{V}_{m}}^{p+1}).Since (u(τ−t),v(τ−t))∈B0(τ−t,θ−τw)\left(u\left(\tau -t),v\left(\tau -t))\in {B}_{0}\left(\tau -t,{\theta }_{-\tau }w)when t→+∞t\to +\infty (4.20)e−σt‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx≤C03e−σt(1+‖v(τ−t)‖Lμ22+‖∇mu(τ−t)‖2+‖∇mu(τ−t)‖p+1)→0,\begin{array}{l}{e}^{-\sigma t}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]\\ \hspace{1.0em}\le {C}_{03}{e}^{-\sigma t}\left(1+\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{p+1})\to 0,\end{array}and there exists T0=T0(τ,w,B0){T}_{0}={T}_{0}\left(\tau ,w,{B}_{0})such that for all t≥T0t\ge {T}_{0}, (4.21)C03e−σt(1+‖v(τ−t)‖Lμ22+‖∇mu(τ−t)‖2+‖∇mu(τ−t)‖p+1)≤1.\hspace{-30.4em}{C}_{03}{e}^{-\sigma t}\left(1+\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{p+1})\le 1.By (3.4), it is easy to get to any t≥0t\ge 0, (4.22)−2∫ΩG(x,u)dx≤−2c3∫Ω∣u∣p+1dx+2∫Ωϕ3dx≤2∫Ωϕ3dx.\hspace{-30.4em}-2\mathop{\int }\limits_{\Omega }G\left(x,u){\rm{d}}x\le -2{c}_{3}\mathop{\int }\limits_{\Omega }| u{| }^{p+1}{\rm{d}}x+2\mathop{\int }\limits_{\Omega }{\phi }_{3}{\rm{d}}x\le 2\mathop{\int }\limits_{\Omega }{\phi }_{3}{\rm{d}}x.When ∣ξ∣→∞| \xi | \to \infty , w(ξ)w\left(\xi )at most polynomial growth, C01e−στ∫−∞τeσξ(1+∣θ−τw(ξ)∣2+∣θ−τw(ξ)∣21−2q+∣θ−τw(ξ)∣p+1)dξ≡r00(τ,w).{C}_{01}{e}^{-\sigma \tau }\underset{-\infty }{\overset{\tau }{\int }}{e}^{\sigma \xi }\left(1+| {\theta }_{-\tau }w\left(\xi ){| }^{2}+| {\theta }_{-\tau }w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| {\theta }_{-\tau }w\left(\xi ){| }^{p+1}){\rm{d}}\xi \equiv {r}_{00}\left(\tau ,w).We get from (4.18) and (4.21) that (4.23)‖v(τ,τ−t,θ−τw,vτ−t)‖Lμ22+‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2≤C041+2α−1λ11−me−στ∫−∞τeσξ‖f(⋅,ξ)‖Lμ22dξ+r00(τ,w)≡r10(τ,w),\begin{array}{l}\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}\le {C}_{04}\left(1+2{\alpha }^{-1}{\lambda }_{1}^{1-m}{e}^{-\sigma \tau }\underset{-\infty }{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \right)+{r}_{00}\left(\tau ,w)\equiv {r}_{10}\left(\tau ,w),\end{array}and r10(τ,w){r}_{10}\left(\tau ,w)is bounded.□Taking the inner product of (3.8) with (−Δ)kv,k=1,2,…,m−1{\left(-\Delta )}^{k}v,k=1,2,\ldots ,m-1in L2(Ω){L}^{2}\left(\Omega ), we find that (4.24)12ddt‖∇kv‖2=ε‖∇kv‖2−(a(x)(−Δ)mv,(−Δ)kv)+ε(a(x)(−Δ)mu,(−Δ)kv)−ε2(u,(−Δ)kv)−(b(x)M(‖∇mu‖2)(−Δ)mu,(−Δ)kv)−(g(x,u),(−Δ)kv)+(f(x,t),(−Δ)kv)+εw(t)(h,(−Δ)kv)−w(t)(a(x)(−Δ)mh,(−Δ)kv).\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}v{\Vert }^{2}& =& \varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}-\left(a\left(x){\left(-\Delta )}^{m}v,{\left(-\Delta )}^{k}v)+\varepsilon \left(a\left(x){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)\\ & & \phantom{\rule[-0.75em]{}{0ex}}-{\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}v)-\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)-\left(g\left(x,u),{\left(-\Delta )}^{k}v)\\ & & +(f\left(x,t),{\left(-\Delta )}^{k}v)+\varepsilon w\left(t)\left(h,{\left(-\Delta )}^{k}v)-w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v).\end{array}For each term on the right-hand side of (4.24): (4.25)(a(x)(−Δ)mv,(−Δ)kv)=(a(x)∇m+kv,∇m+kv)+∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+kv,\left(a\left(x){\left(-\Delta )}^{m}v,{\left(-\Delta )}^{k}v)=\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)+\left(\mathop{\sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v\right),(4.26)ε(a(x)(−Δ)mu,(−Δ)kv)=ε(a(x)∇m+ku,∇m+kv)+ε∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+ku=ε12ddt(a(x)∇m+ku,∇m+ku)+ε2(a(x)∇m+ku,∇m+ku)−εw(t)(a(x)∇m+ku,∇m+kh)+ε∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+ku,\hspace{2.35em}\begin{array}{l}\varepsilon \left(a\left(x){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)=\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}v)+\varepsilon \left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u\right)\\ \hspace{9.82em}=\hspace{-0.25em}\varepsilon \frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{10.82em}-\hspace{0.33em}\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+\hspace{-0.25em}\varepsilon \left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u\right),\end{array}(4.27)ε2(u,(−Δ)kv)=ε2(u,(−Δ)k(ut+εu−hw(t)))=ε22ddt‖∇ku‖2+ε3‖∇ku‖2−ε2w(t)(∇ku,∇kh),{\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}v)={\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}\left({u}_{t}+\varepsilon u-hw\left(t)))=\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}-{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h),(4.28)(b(x)M(‖∇mu‖2)(−Δ)mu,(−Δ)kv)=b0M(‖∇mu‖2)(a(x)∇m+ku,∇m+kv)+M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku=b02M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−b0M(‖∇mu‖2)w(t)(a(x)∇m+ku,∇m+kh)+M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku,\hspace{-39.1em}\begin{array}{l}\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)\\ \hspace{1.0em}={b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}v)+M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right)\\ \hspace{1.0em}=\frac{{b}_{0}}{2}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{2.0em}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right),\end{array}(4.29)(g(x,u),(−Δ)kv)=(∇xkg(x,u),∇kv)≤∫Ω(c5∣u∣p+ϕ5(x))∇kvdx≤c5∫Ω∣u∣p∇kvdx+∫Ωϕ5(x)∇kvdx≤c5‖u‖L2pp‖∇kv‖+‖ϕ5(x)‖‖∇kv‖≤a008‖∇m+kv‖2+Ck1(r01p(τ,w)+‖ϕ5(x)‖2),\hspace{-39.1em}\begin{array}{rcl}\left(g\left(x,u),{\left(-\Delta )}^{k}v)& =& \left({\nabla }_{x}^{k}g\left(x,u),{\nabla }^{k}v)\le \left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }\left({c}_{5}| u{| }^{p}+{\phi }_{5}\left(x)){\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|\\ & \le & {c}_{5}\left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }| u{| }^{p}{\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|+\left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }{\phi }_{5}\left(x){\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|\\ & \le & {c}_{5}\Vert u{\Vert }_{{L}^{2p}}^{p}\Vert {\nabla }^{k}v\Vert +\Vert {\phi }_{5}\left(x)\Vert \Vert {\nabla }^{k}v\Vert \\ & \le & \frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}+{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2}),\end{array}(4.30)(f(x,t),(−Δ)kv)=(∇kf(x,t),∇kv)≤‖∇kf(x,t)‖‖∇kv‖≤a008‖∇m+kv‖2+2λ1−ma00‖∇kf(x,t)‖2,\hspace{-38.92em}(f\left(x,t),{\left(-\Delta )}^{k}v)=\left({\nabla }^{k}f\left(x,t),{\nabla }^{k}v)\le \Vert {\nabla }^{k}f\left(x,t)\Vert \Vert {\nabla }^{k}v\Vert \le \frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2},moreover, (4.31)(∇m+k−iv∇ia(x),∇m+kv)≤ai‖∇m+k−iv‖‖∇m+kv‖,i=1,2,…,m−k,ai=‖∇ia(x)‖∞.\left({\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v)\le {a}_{i}\Vert {\nabla }^{m+k-i}v\Vert \Vert {\nabla }^{m+k}v\Vert ,\hspace{1em}i=1,2,\ldots ,m-k,{a}_{i}=\Vert {\nabla }^{i}a\left(x){\Vert }_{\infty }.According to the interpolation inequality, we have ‖∇m+k−iv‖≤Ci‖∇m+kv‖αi‖v‖1−αi,αi=m+k−im+k,\Vert {\nabla }^{m+k-i}v\Vert \le {C}_{i}\Vert {\nabla }^{m+k}v{\Vert }^{{\alpha }_{i}}\Vert v{\Vert }^{1-{\alpha }_{i}},\hspace{1em}{\alpha }_{i}=\frac{m+k-i}{m+k},then (4.32)(Cm−ki∇m+k−iv∇ia(x),∇m+kv)≤Cm−kiCiai‖v‖1−αi‖∇m+kv‖1+αi≤a008(m−k)‖∇m+kv‖2+1−αi2a004(1−αi)(m−k)1+αi1−αi(Cm−kiCiai)21−αi‖v‖2,\begin{array}{lcl}\left({C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v)& \le & {C}_{m-k}^{i}{C}_{i}{a}_{i}\Vert v{\Vert }^{1-{\alpha }_{i}}\Vert {\nabla }^{m+k}v{\Vert }^{1+{\alpha }_{i}}\\ & \le & \frac{{a}_{00}}{8\left(m-k)}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\frac{1-{\alpha }_{i}}{2}{\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{C}_{i}{a}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2},\end{array}(4.33)(Cm−ki∇m+k−iv∇ia(x),∇m+ku)≤a00b0M08(m−k)‖∇m+ku‖2+2(m−k)a00b0M0(Cm−kiai)2‖∇m+k−iv‖2≤a00b0M08(m−k)‖∇m+ku‖2+a008(m−k)‖∇m+kv‖2+(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2,\hspace{-41.2em}\begin{array}{lcl}\left({C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u)& \le & \frac{{a}_{00}{b}_{0}{M}_{0}}{8\left(m-k)}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i})}^{2}\Vert {\nabla }^{m+k-i}v{\Vert }^{2}\\ & \hspace{0.425em}\le \hspace{-0.25em}& \frac{{a}_{00}{b}_{0}{M}_{0}}{8\left(m-k)}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{{a}_{00}}{8\left(m-k)}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2},\end{array}(4.34)b0M(‖∇mu‖2)w(t)(a(x)∇m+ku,∇m+kh)≤b0a0M0(1+‖∇mu‖2q)∣w(t)∣‖∇m+ku‖‖∇m+kh‖≤εb0a00M08‖∇m+ku‖2+2ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2,\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)\\ \hspace{1.0em}\le {b}_{0}{a}_{0}{M}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})| w\left(t)| \Vert {\nabla }^{m+k}u\Vert \Vert {\nabla }^{m+k}h\Vert \\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+2{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2},\end{array}(4.35)M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku≤M0b0(1+‖∇mu‖2q)∑i=1m−kCm−kiai‖∇m+k−iv‖‖∇m+ku‖≤εb0a00M08‖∇m+ku‖2+ε−1a00−1b0M0∑i=1m−k(Cm−ki(1+‖∇mu‖2q)ai)2‖∇m+k−iv‖2≤εb0a00M08‖∇m+ku‖2+a008‖∇m+kv‖2+(ε−1a00−1b0M0)11−αi∑i=1m−k(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2.\begin{array}{l}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right)\\ \hspace{1.0em}\le {M}_{0}{b}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{a}_{i}\Vert {\nabla }^{m+k-i}v\Vert \Vert {\nabla }^{m+k}u\Vert \\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}){a}_{i})}^{2}\Vert {\nabla }^{m+k-i}v{\Vert }^{2}\\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}\\ \hspace{2.0em}+\hspace{0.25em}{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}.\end{array}By (4.25)–(4.30) and (4.32)–(4.35), we get (4.36)ddt[‖∇kv‖2−ε(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2(a(x)∇m+kv,∇m+kv)−4+ε4a00‖∇m+kv‖2−2ε‖∇kv‖2+2εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)+2ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}-\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]+{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{4+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-2\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\end{array}+2∑i=1m−k(ε−1a00−1b0M0)11−αi(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)−2εw(t)(a(x)∇m+ku,∇m+kh)+2ε2w(t)(∇ku,∇kh)+2εw(t)((−Δ)ku,h)−2w(t)(a(x)(−Δ)mh,(−Δ)kv).\begin{array}{l}\hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})-2\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+2{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h)\\ \hspace{2.0em}+2\varepsilon w\left(t)\left({\left(-\Delta )}^{k}u,h)-2w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v).\end{array}Using the Cauchy-Schwarz inequality,Young’s inequality and Holder’s inequality, etc. we have (4.37)2εw(t)(a(x)∇m+ku,∇m+kh)≤ε2a00‖∇m+ku‖2+a00−1a02∣w(t)∣2‖∇m+kh‖2,2\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)\le {\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2},(4.38)2ε2w(t)(∇ku,∇kh)≤ε3‖∇ku‖2+ε∣w(t)∣2‖∇kh‖2,\hspace{-21.6em}2{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h)\le {\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2},(4.39)2εw(t)((−Δ)ku,h)≤ε‖∇kv‖2+ε∣w(t)∣2‖∇kh‖2,\hspace{-21.6em}2\varepsilon w\left(t)\left({\left(-\Delta )}^{k}u,h)\le \varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2},(4.40)2w(t)(a(x)(−Δ)mh,(−Δ)kv)=2w(t)(a(x)∇m+kv,∇m+kh)+2w(t)∑i=1m−kCm−ki∇m+k−i∇ia(x),∇m+kh≤a004‖∇m+kv‖2+4a00−1a02∣w(t)∣2‖∇m+k‖2+a004‖∇m+kv‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2≤a002‖∇m+kv‖2+4a00−1a02∣w(t)∣2‖∇m+k‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}2w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v)\\ \hspace{1.0em}=2w\left(t)\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}h)+2w\left(t)\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}{\nabla }^{i}a\left(x),{\nabla }^{m+k}h\right)\\ \hspace{1.0em}\le \frac{{a}_{00}}{4}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+\frac{{a}_{00}}{4}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}\\ \hspace{1.0em}\le \frac{{a}_{00}}{2}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}Substitute (4.37)–(4.40) into (4.36) to obtain (4.41)ddt[‖∇kv‖2−ε(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2(a(x)∇m+kv,∇m+kv)−6+ε4a00‖∇m+kv‖2−3ε‖∇kv‖2+2εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)−ε2a00‖∇m+ku‖2+ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2+2∑i=1m−k(1−αi)(ε−1a00−1b0M0)11−αia008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+k‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}-\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]+{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}{\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}When ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge 0, b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥ddt(b0M0(a(x)∇m+ku,∇m+ku))+2εb0M0(a(x)∇m+ku,∇m+ku);\hspace{-34em}\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\right)\\ \hspace{1.0em}\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}{M}_{0}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u))+2\varepsilon {b}_{0}{M}_{0}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u);\end{array}else b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥ddt(b0C(‖∇mu‖2)(a(x)∇m+ku,∇m+ku))+2εb0C(‖∇mu‖2)(a(x)∇m+ku,∇m+ku),\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\right)\\ \hspace{1.0em}\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u))+2\varepsilon {b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u),\end{array}then (4.41) is transformed into (4.42)ddt[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+2(a(x)∇m+kv,∇m+kv)−6+ε4a00‖∇m+kv‖2−3ε‖∇kv‖2+2εb0M0(C(‖∇mu‖2))(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)−ε2a00‖∇m+ku‖2+ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2+2∑i=1m−k(1−αi)(ε−1a00−1b0M0)11−αia008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+k‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}{\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}By 2(a(x)∇m+kv,∇m+kv)≥2a00‖∇m+kv‖2,(a(x)∇m+ku,∇m+ku)≥a00‖∇m+ku‖22\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)\ge 2{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2},\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge {a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}, (4.1), we have (4.43)ddt[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+σ1[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]≤Ck2(1+∣w(t)∣2)+2λ1−ma00‖∇kf(x,t)‖2.\hspace{-34.3em}\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1em}+{\sigma }_{1}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1em}\le {C}_{k2}\left(1+| w\left(t){| }^{2})+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}.\end{array}When k=mk=m, (4.43) is also true, which will not be detailed here.Using the Gronwall inequality to integrate (4.43) over [τ−t,τ]\left[\tau -t,\tau ]and replacing wwby θ−τw{\theta }_{-\tau }wwe obtain (4.44)eσ1τ[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]≤eσ1(τ−t)[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]+∫τ−tτeσ1ξCk2(1+∣w(ξ)∣2)dξ+2λ1−ma00∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ,\begin{array}{l}{e}^{{\sigma }_{1}\tau }\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1.0em}\le {e}^{{\sigma }_{1}\left(\tau -t)}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]+\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi ,\end{array}moreover, (4.45)‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2≤e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk2(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi .\end{array}Since (u(τ−t),v(τ−t))∈Bk(τ−t,θ−τw)\left(u\left(\tau -t),v\left(\tau -t))\in {B}_{k}\left(\tau -t,{\theta }_{-\tau }w), when t→+∞t\to +\infty , (4.46)e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]≤e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)a0‖∇m+ku(τ−t)‖2+ε2‖∇ku(τ−t)‖2]→0,\begin{array}{l}{e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon ){a}_{0}\Vert {\nabla }^{m+k}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\to 0,\end{array}then there exists Tk=Tk(τ,w,Bk){T}_{k}={T}_{k}\left(\tau ,w,{B}_{k})such that for all t≥Tkt\ge {T}_{k}(4.47)e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)a0‖∇m+ku(τ−t)‖2+ε2‖∇ku(τ−t)‖2]≤1.{e}^{-{\sigma }_{1}t}\left[\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon ){a}_{0}\Vert {\nabla }^{m+k}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\le 1.When ∣ξ∣→∞w(ξ)| \xi | \to \infty \hspace{0.33em}w\left(\xi )at most polynomial growth, e−σ1τ∫−∞τeσ1ξCk2(1+∣w(ξ)∣2)dξ≡r0k(τ,w).{e}^{-{\sigma }_{1}\tau }\underset{-\infty }{\overset{\tau }{\int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi \equiv {r}_{0k}\left(\tau ,w).We get from (4.45)–(4.47), (a(x)∇m+ku,∇m+ku)≥a00‖∇m+ku‖2\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge {a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}that (4.48)‖∇kv(τ,τ−t,θ−τw,vτ−t)‖2+‖∇m+ku(τ,τ−t,θ−τw,vτ−t)‖2≤Ck31+2λ1−ma00e−σ1τ∫−∞τeσ1ξ‖∇kf(x,ξ)‖2dξ+r0k(τ,w)≡r1k(τ,w),\begin{array}{l}\Vert {\nabla }^{k}v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}+\Vert {\nabla }^{m+k}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}\le {C}_{k3}\left(1+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{-\infty }{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi \right)+{r}_{0k}\left(\tau ,w)\equiv {r}_{1k}\left(\tau ,w),\end{array}and r1k(τ,w){r}_{1k}\left(\tau ,w)are bounded.Lemma 4.1 is derived from (4.23) and (4.48).Lemma 4.1 is proved.Considering the eigenvalue problem (−Δ)m+ku=λm+ku,u∣Γ=0,{\left(-\Delta )}^{m+k}u={\lambda }^{m+k}u,u\hspace{-0.25em}{| }_{\Gamma }=0,the problem (3.8) has a family of eigenfunctions {ej}j=1∞{\left\{{e}_{j}\right\}}_{j=1}^{\infty }with the eigenvalues {λj}j=1∞:λ1≤λ2≤⋯≤λj→∞(j→∞){\left\{{\lambda }_{j}\right\}}_{j=1}^{\infty }:{\lambda }_{1}\le {\lambda }_{2}\hspace{0.33em}\le \cdots \le {\lambda }_{j}\to \infty \left(j\to \infty ), such that {ej}j=1∞{\left\{{e}_{j}\right\}}_{j=1}^{\infty }is an orthonormal basis of Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ). Given nnlet Qn=span{e1,⋅,en}{Q}_{n}={\rm{span}}\left\{{e}_{1},\cdot ,{e}_{n}\right\}and Pn:Vk(Ω)→Qn{P}_{n}\hspace{0.25em}:{V}_{k}\left(\Omega )\to {Q}_{n}be the projection operator.Lemma 4.2Suppose MMsatisfies (M),h(x)∈Vm+k(Ω),k=0,1,…,m\left(M),h\left(x)\in {V}_{m+k}\left(\Omega ),k=0,1,\ldots ,m(3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1})and for ∀η>0,τ∈R,w∈Ω1,Bk={Bk(τ,w):τ∈R,w∈Ω1}∈Dk\forall \hspace{-0.25em}\eta \gt 0,\tau \in R,w\in {\Omega }_{1},{B}_{k}=\left\{{B}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}, there exists Tk=Tk(τ,w,Bk,ηk)>0,Nk=Nk(τ,w,ηk)≥0{T}_{k}={T}_{k}\left(\tau ,w,{B}_{k},{\eta }_{k})\gt 0,{N}_{k}={N}_{k}\left(\tau ,w,{\eta }_{k})\ge 0such that the solution of (3.8) satisfies for t≥Tk,n≥Nkt\ge {T}_{k},n\ge {N}_{k}‖(I−Pn)v(τ,τ−t,θ−τw)‖Vk2+‖(I−Pn)u(τ,τ−t,θ−τw)‖Vm+k2≤ηk.\Vert \left(I-{P}_{n})v\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{V}_{k}}^{2}+\Vert \left(I-{P}_{n})u\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{V}_{m+k}}^{2}\le {\eta }_{k}.ProofLet un,1=Pnu,un,2=u−un,1,vn,1=Pnv,vn,2=v−vn,1{u}_{n,1}={P}_{n}u,{u}_{n,2}=u-{u}_{n,1},{v}_{n,1}={P}_{n}v,{v}_{n,2}=v-{v}_{n,1}. Applying (I−Pn)\left(I-{P}_{n})to the second equation of (3.8), we obtain (4.49)dvn,2dt=εvn,2−a(x)(−Δ)mvn,2+εa(x)(−Δ)mun,2−ε2un,2−b(x)(I−Pn)M(‖∇m‖2)(−Δ)mu−(I−Pn)(g(x,u)+f(x,t))+εh(x)w(t)−a(x)(−Δ)mh(x)w(t).\begin{array}{l}\frac{{\rm{d}}{v}_{n,2}}{{\rm{d}}t}=\varepsilon {v}_{n,2}-a\left(x){\left(-\Delta )}^{m}{v}_{n,2}+\varepsilon a\left(x){\left(-\Delta )}^{m}{u}_{n,2}-{\varepsilon }^{2}{u}_{n,2}-b\left(x)\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}{\Vert }^{2}){\left(-\Delta )}^{m}u\\ \hspace{3.25em}-\left(I-{P}_{n})\left(g\left(x,u)+f\left(x,t))+\varepsilon h\left(x)w\left(t)-a\left(x){\left(-\Delta )}^{m}h\left(x)w\left(t).\end{array}Taking the inner product of the resulting equation (4.49) with vn,2{v}_{n,2}in Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ), we have (4.50)12ddt‖vn,2‖Lμ22=ε‖vn,2‖Lμ22−‖∇mvn,2‖2+ε((−Δ)mun,2,vn,2)−ε2(un,2,vn,2)Lμ2−b0((I−Pn)M(‖∇mu‖2)(−Δ)mu,vn,2)−((I−Pn)g(x,u),vn,2)Lμ2+((I−Pn)f(x,t),vn,2)Lμ2+εw(t)(h(x),vn,2)Lμ2−w(t)((−Δ)mh,vn,2).\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}& =& \varepsilon \Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}-\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},{v}_{n,2})-{\varepsilon }^{2}{\left({u}_{n,2},{v}_{n,2})}_{{L}_{\mu }^{2}}\\ & & -{b}_{0}\left(\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{v}_{n,2})-{\left(\left(I-{P}_{n})g\left(x,u),{v}_{n,2})}_{{L}_{\mu }^{2}}\\ & & +{\left(\left(I-{P}_{n})f\left(x,t),{v}_{n,2})}_{{L}_{\mu }^{2}}+\varepsilon w\left(t){\left(h\left(x),{v}_{n,2})}_{{L}_{\mu }^{2}}-w\left(t)\left({\left(-\Delta )}^{m}h,{v}_{n,2}).\end{array}Applying I−PnI-{P}_{n}to the first equation of (3.8), we get (4.51)vn,2=dun,2dt+εun,2−hw(t).{v}_{n,2}=\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t).For the third and fourth terms on the right-hand side of (4.49), we obtain (4.52)ε((−Δ)mun,2,vn,2)−ε2(un,2,vn,2)Lμ2=ε((−Δ)mun,2,dun,2dt+εun,2−hw(t))−ε2(un,2,dun,2dt+εun,2−hw(t))Lμ2=ε2ddt‖∇mun,2‖2+ε2‖∇mun,2‖2−εw(t)((−Δ)mun,2,h)−ε22ddt‖un,2‖Lμ22−ε3‖un,2‖Lμ22+ε2w(t)(un,2,h)Lμ2,\begin{array}{l}\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},{v}_{n,2})-{\varepsilon }^{2}{\left({u}_{n,2},{v}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))-\hspace{-0.25em}{\varepsilon }^{2}{\left({u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\hspace{0.25em}\frac{\varepsilon }{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}-\varepsilon w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h)\\ \hspace{2.0em}-\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}-{\varepsilon }^{3}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{2}w\left(t){\left({u}_{n,2},h)}_{{L}_{\mu }^{2}},\end{array}for the fifth term on the right-hand side of (4.49), we have (4.53)b0((I−Pn)M(‖∇mu‖2)(−Δ)mu,vn,2)=b0(M(‖∇mu‖2)(−Δ)mun,2,dun,2dt+εun,2−hw(t))=12b0M(‖∇mu‖2)ddt‖∇mun,2‖2+εb0M(‖∇mu‖2)∣∇mun,2‖2−b0M(‖∇mu‖2)w(t)((−Δ)mun,2,h),\begin{array}{l}{b}_{0}\left(\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{v}_{n,2})\\ \hspace{1.0em}={b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}{u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))\\ \hspace{1.0em}=\frac{1}{2}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})| {\nabla }^{m}{u}_{n,2}{\Vert }^{2}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h),\end{array}for the sixth term on the right-hand side of (4.49), we have (4.54)((I−Pn)g(x,u),vn,2)Lμ2=((I−Pn)g(x,u),dun,2dt+εun,2−hw(t))Lμ2=ddt((I−Pn)g(x,u),un,2)Lμ2−((I−Pn)gu(x,u)ut,un,2)Lμ2+ε((I−Pn)g(x,u),un,2)Lμ2−((I−Pn)g(x,u),hw(t))Lμ2,\begin{array}{l}{\left(\left(I-{P}_{n})g\left(x,u),{v}_{n,2})}_{{L}_{\mu }^{2}}=\hspace{-0.25em}{\left(\left(I-{P}_{n})g\left(x,u),\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\frac{{\rm{d}}}{{\rm{d}}t}{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}-{\left(\left(I-{P}_{n}){g}_{u}\left(x,u){u}_{t},{u}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+\hspace{0.25em}\varepsilon {\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}-{\left(\left(I-{P}_{n})g\left(x,u),hw\left(t))}_{{L}_{\mu }^{2}},\end{array}for the seventh, eighth, and ninth terms on the right-hand side of (4.49), we have (4.55)((I−Pn)f(x,t),vn,2)Lμ2≤14‖∇mvn,2‖2+1αn+1λn+1m−1‖(I−Pn)f(x,t)‖Lμ22,{\left(\left(I-{P}_{n})f\left(x,t),{v}_{n,2})}_{{L}_{\mu }^{2}}\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\frac{1}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{P}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2},(4.56)εw(t)(h(x),vn,2)Lμ2≤14‖∇mvn,2‖2+ε∣w(t)∣2αn+1λn+1m−1‖h(x)‖Lμ22,\hspace{-24.5em}\varepsilon w\left(t){\left(h\left(x),{v}_{n,2})}_{{L}_{\mu }^{2}}\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\frac{\varepsilon | w\left(t){| }^{2}}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2},(4.57)w(t)((−Δ)mh,vn,2)≤14‖∇mvn,2‖2+ε∣w(t)∣2‖∇mh(x)‖2.\hspace{-24.5em}w\left(t)\left({\left(-\Delta )}^{m}h,{v}_{n,2})\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{m}h\left(x){\Vert }^{2}.When ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥0\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\ge 0(4.58)b0M(‖∇mu‖2)ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥ddt(b0M0‖∇mun,2‖2)+2εb0M0‖∇mvn,2‖2,{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\right)\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}{M}_{0}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2})+2\varepsilon {b}_{0}{M}_{0}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2},else (4.59)b0M(‖∇mu‖2)ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥ddt(b0C(‖∇mu‖2)‖∇mun,2‖2)+2εb0C(‖∇mu‖2)‖∇mvn,2‖2.{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\right)\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2})+2\varepsilon {b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}.By using Young’s inequality and Holder’s inequality, we can get (4.60)2b0M(‖∇mu‖2)w(t)((−Δ)mun,2,h)≤2b0M(‖∇mu‖2)w(t)‖∇mun,2‖∣w(t)∣‖∇mh‖≤εb0M02‖∇mun,2‖2+2b0C(‖∇mu‖2)εM0∣w(t)∣2‖∇mh‖2,\begin{array}{lcl}2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h)& \le \hspace{-0.25em}& 2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\Vert {\nabla }^{m}{u}_{n,2}\Vert | w\left(t)| \Vert {\nabla }^{m}h\Vert \\ & \le & \frac{\varepsilon {b}_{0}{M}_{0}}{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2},\end{array}(4.61)2ε2w(t)(h(x),un,2)Lμ2−2εw(t)((−Δ)mun,2,h(x))≤ε3‖un,2‖Lμ22+ε2‖∇mun,2‖2+ε∣w(t)∣2‖h‖Lμ22+∣w(t)∣2‖∇mh‖2.\hspace{-35.35em}\begin{array}{l}2{\varepsilon }^{2}w\left(t){\left(h\left(x),{u}_{n,2})}_{{L}_{\mu }^{2}}-2\varepsilon w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h\left(x))\\ \hspace{1.0em}\le {\varepsilon }^{3}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}.\end{array}By (3.5), we have (4.62)2((I−Pn)gu(x,u)ut,un,2)Lμ2≤2‖ϕ4‖Lμ4‖ut‖Lμ2‖un,2‖Lμ4+2c4‖ut‖Lμ2‖u‖Lμ2pp−1‖un,2‖Lμ2p≤εb0M02‖∇mun,2‖2+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2,\begin{array}{l}2{\left(\left(I-{P}_{n}){g}_{u}\left(x,u){u}_{t},{u}_{n,2})}_{{L}_{\mu }^{2}}\le \hspace{-0.25em}2\Vert {\phi }_{4}{\Vert }_{{L}_{\mu }^{4}}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{4}}+2{c}_{4}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}\Vert u{\Vert }_{{L}_{\mu }^{2p}}^{p-1}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2p}}\le \hspace{-0.25em}\frac{\varepsilon {b}_{0}{M}_{0}}{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2},\end{array}(4.63)2w(t)((I−Pn)g(x,u),h)Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣‖u‖Lμ2pp‖h‖Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2C07∣w(t)∣‖∇mu‖p‖h‖Lμ2.\hspace{-38.78em}\begin{array}{l}2w\left(t){\left(\left(I-{P}_{n})g\left(x,u),h)}_{{L}_{\mu }^{2}}\le \hspace{-0.25em}2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| \Vert u{\Vert }_{{L}_{\mu }^{2p}}^{p}\Vert h{\Vert }_{{L}_{\mu }^{2}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{07}| w\left(t)| \Vert {\nabla }^{m}u{\Vert }^{p}\Vert h{\Vert }_{{L}_{\mu }^{2}}.\end{array}By substituting (4.52)–(4.63) into (4.50), we have (4.64)ddt[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]+12‖∇mvn,2‖2−2ε‖vn,2‖Lμ22+(2εM(‖∇mu‖2)−εM0)b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+2ε((I−Pn)g(x,u),un,2)Lμ2≤1+2b0C(‖∇mu‖2)εM0+2ε∣w(t)∣2‖∇mh‖2+ε+2εαn+1λn+1m−1∣w(t)∣2‖h(x)‖Lμ22+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2+(2‖ϕ1‖Lμ2‖h‖Lμ2+2C08‖∇mu‖p)∣w(t)∣‖h‖Lμ2+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}+\frac{1}{2}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}-2\varepsilon \Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\varepsilon {M}_{0}){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}\\ \hspace{1.0em}-\hspace{0.25em}3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le \hspace{-0.25em}\left(1+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}+2\varepsilon \right)| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\left(\varepsilon +\frac{2\varepsilon }{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\right)| w\left(t){| }^{2}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\hspace{0.25em}\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+\hspace{0.25em}{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2}+\left(2\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{08}\Vert {\nabla }^{m}u{\Vert }^{p})| w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}\hspace{2em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}.\end{array}Because, when N=1,2,then1≤p<+∞N=1,2,\text{then}\hspace{0.25em}1\le p\lt +\infty ; when N=3,4,then1≤p<NN−2N=3,4,\text{then}\hspace{0.25em}1\le p\lt \frac{N}{N-2}. Moreover, when n→∞,λn→∞n\to \infty ,{\lambda }_{n}\to \infty so given η0>0{\eta }_{0}\gt 0, there exists N01=N01(η0)≥1{N}_{01}={N}_{01}\left({\eta }_{0})\ge 1for n≥N01n\ge {N}_{01}(4.65)1+2b0C(‖∇mu‖2)εM0+2ε∣w(t)∣2‖∇mh‖2+ε+2εαn+1λn+1m−1∣w(t)∣2‖h(x)‖Lμ22+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2+(2‖ϕ1‖Lμ2‖h‖Lμ2+2C08‖∇mu‖p)∣w(t)∣‖h‖Lμ2+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22≤C09η0(1+∣w(t)∣2+‖ut‖Lμ26+‖∇mu‖6)+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22.\begin{array}{l}\left(1+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}+2\varepsilon \right)| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\left(\varepsilon +\frac{2\varepsilon }{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\right)| w\left(t){| }^{2}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2}\\ \hspace{1.0em}+\left(2\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{08}\Vert {\nabla }^{m}u{\Vert }^{p})| w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}\le {C}_{09}{\eta }_{0}\left(1+| w\left(t){| }^{2}+\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u{\Vert }^{6})+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}.\end{array}Then, there is an appropriate positive constant σ2{\sigma }_{2}so that (4.64) can be reduced to (4.66)ddt[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]+σ2[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]≤C09η0(1+∣w(t)∣2+‖ut‖Lμ26+‖∇mu‖6)+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22,\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}+{\sigma }_{2}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}\le {C}_{09}{\eta }_{0}\left(1+| w\left(t){| }^{2}+\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u{\Vert }^{6})+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2},\end{array}integrating (4.66) over (τ−t,τ)\left(\tau -t,\tau )with t≥0t\ge 0we get for all n≥N01n\ge {N}_{01}(4.67)‖vn,2(τ,τ−t,w)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,w)‖2+ε2‖un,2(τ,τ−t,w)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,w))Lμ2\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,w){\Vert }^{2}\\ \hspace{1.0em}+\hspace{0.25em}{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,w))}_{{L}_{\mu }^{2}}\end{array}≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0∫τ−tτeσ(s−t)(1+∣w(s)∣2+‖ut(s,τ−t,w,u1τ)‖Lμ26+‖∇mu(s,τ−t,w,u1τ)‖6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds.\begin{array}{l}\hspace{1.0em}\le \hspace{-0.25em}{e}^{-{\sigma }_{2}t}\left[\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}\hspace{2em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\hspace{2em}+{C}_{09}{\eta }_{0}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\left(1+| w\left(s){| }^{2}+\Vert {u}_{t}\left(s,\tau -t,w,{u}_{1\tau }){\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u\left(s,\tau -t,w,{u}_{1\tau }){\Vert }^{6}){\rm{d}}s+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}s.\end{array}Replacing wwby θ−τw{\theta }_{-\tau }win (4.67) for every t∈R+,τ∈R,w∈Ω1,n≥N01t\in {R}^{+},\tau \in R,w\in {\Omega }_{1},n\ge {N}_{01}, we obtain (4.68)‖vn,2(τ,τ−t,θ−τw)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,θ−τw)‖2+ε2‖un,2(τ,τ−t,θ−τw)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,θ−τw))Lμ2≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0∫τ−tτeσ(s−t)(1+∣w(s)∣2+‖ut(s,τ−t,θ−τw,u1τ)‖Lμ26+‖∇mu(s,τ−t,θ−τw,u1τ)‖6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds.\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}\\ \hspace{1.0em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{2}t}{[}\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}\hspace{2em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\hspace{2em}+{C}_{09}{\eta }_{0}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}(1+| w\left(s){| }^{2}+\Vert {u}_{t}\left(s,\tau -t,{\theta }_{-\tau }w,{u}_{1\tau }){\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u\left(s,\tau -t,{\theta }_{-\tau }w,{u}_{1\tau }){\Vert }^{6}){\rm{d}}s\\ \hspace{2.0em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}s.\end{array}From (3.8), f(x,t)f\left(x,t)satisfies (F1),h∈Vm\left({F}_{1}),h\in {V}_{m}and Lemma 4.1, we have (4.69)‖vn,2(τ,τ−t,θ−τw)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,θ−τw)‖2+ε2‖un,2(τ,τ−t,θ−τw)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,θ−τw))Lμ2≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0r003∫τ−tτeσ(s−t)(1+∣w(s)∣2+∣w(s)∣6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds,\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{2}t}\left[\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]+{C}_{09}{\eta }_{0}{r}_{00}^{3}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\left(1+| w\left(s){| }^{2}+| w\left(s){| }^{6}){\rm{d}}s\\ \hspace{2.0em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}\hspace{0.33em}{\rm{d}}s,\end{array}by (uτ−t,vτ−t)∈B0(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{0}\left(\tau -t,{\theta }_{-\tau }w), then (4.70)e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]→0,t→∞.\begin{array}{l}{e}^{-{\sigma }_{2}t}{[}\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\to 0,\hspace{1em}t\to \infty .\end{array}Taking the inner product of (4.49) with (−Δ)kvn,2,k=1,2,…,m−1{\left(-\Delta )}^{k}{v}_{n,2},k=1,2,\ldots ,m-1in L2(Ω){L}^{2}\left(\Omega ), we have (4.71)12ddt‖∇kvn,2‖2=ε‖∇kvn,2‖2−(a(x)(−Δ)mvn,2,(−Δ)kvn,2)+ε(a(x)(−Δ)mun,2,(−Δ)kvn,2)−ε2(un,2,(−Δ)kvn,2)−(b(x)M(‖∇mu‖2)(−Δ)mun,2,(−Δ)kvn,2)−((I−Pn)g(x,u),(−Δ)kvn,2)+(f(x,t),(−Δ)kvn,2)+εw(t)(h,(−Δ)kvn,2)−w(t)(a(x)(−Δ)mh,(−Δ)kvn,2).\begin{array}{l}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}=\hspace{-0.25em}\varepsilon \Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}-\left(a\left(x){\left(-\Delta )}^{m}{v}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})+\varepsilon \left(a\left(x){\left(-\Delta )}^{m}{u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})\\ \hspace{6.67em}-{\varepsilon }^{2}\left({u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})-\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}{u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})\hspace{6.67em}-\left(\left(I-{P}_{n})g\left(x,u),{\left(-\Delta )}^{k}{v}_{n,2})+(f\left(x,t),{\left(-\Delta )}^{k}{v}_{n,2})+\varepsilon w\left(t)\left(h,{\left(-\Delta )}^{k}{v}_{n,2})\hspace{6.67em}-w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}{v}_{n,2}).\end{array}Then applying I−PnI-{P}_{n}to the first equation of (3.8), we obtain (4.72)vn,2=dun,2dt+εun,2−hw(t).{v}_{n,2}=\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t).Combining the processing method of Lemma 4.1, (4.73)ddt[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]+2(a(x)∇m+kvn,2,∇m+kvn,2)−6+ε4a00‖∇m+kvn,2‖2−3ε‖∇kvn,2‖2+2εb0M0(C(‖∇mu‖2))(a(x)∇m+kun,2,∇m+kun,2)−3εa00b0M04‖∇m+kun,2‖2−2ε2(a(x)∇m+kun,2,∇m+kun,2)−ε2a00‖∇m+kun,2‖2+ε3‖∇kun,2‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖vn,2‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖vn,2‖2+2(ε−1a00−1b0M0)11−αi∑i=1m−k(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖vn,2‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λn+1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+kh‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}{v}_{n,2},{\nabla }^{m+k}{v}_{n,2})-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}{v}_{n,2}{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+\hspace{0.25em}2{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{n+1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}Then there is a positive constant σ1{\sigma }_{1}(4.73) can be reduced to (4.74)ddt[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]+σ1[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]≤Ck4(1+∣w(t)∣2)+2λn+1−ma00‖∇kf(x,t)‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}+{\sigma }_{1}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}\le {C}_{k4}\left(1+| w\left(t){| }^{2})+\frac{2{\lambda }_{n+1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}.\end{array}when k=mk=m, (4.74) also holds.□Integrating (4.74) over (τ−t,τ)\left(\tau -t,\tau )with t≥0t\ge 0, we get for all n≥Nk1n\ge {N}_{k1}(4.75)‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2≤e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk4(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k4}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}d\xi .\end{array}Replacing wwby θ−τw{\theta }_{-\tau }win (4.75), we obtain for every t∈R+,τ∈R,w∈Ω1,n≥Nk1t\in {R}^{+},\tau \in R,w\in {\Omega }_{1},n\ge {N}_{k1}(4.76)‖∇kvn,2(τ,τ−t,θ−τw)‖2+(b0M0(C(‖∇mu‖2))−ε)×(a(x)∇m+kun,2(τ,τ−t,θ−τw),∇m+kun,2(τ,τ−t,θ−τw))+ε2‖∇kun,2(τ,τ−t,θ−τw)‖2≤e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk4(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\\ \hspace{1.0em}\times \hspace{0.33em}\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w),{\nabla }^{m+k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k4}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi .\end{array}From (3.8), h∈Vm+kh\in {V}_{m+k}, f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1})Lemma 4.1 and (uτ−t,vτ−t)∈Bk(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{k}\left(\tau -t,{\theta }_{-\tau }w)for t→+∞t\to +\infty (4.77)e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]→0.{e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]\to 0.Combining (4.69), (4.70), (4.76), (4.77), and Lemma 4.1, we can get the conclusion of Lemma 4.2.Lemma 4.2 is proved.5The existence of the family of random attractorsIn this section, we shall prove the existence of the family of random pullback attractors for system (3.8). From Lemma 4.1, we know that for P−a.e.Dk={Dk(τ,w):τ∈R,w∈Ω1}∈DkP-{a.e.}\hspace{0.25em}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}and w∈Ω1w\in {\Omega }_{1}, there exists Tk=Tk(Dk,w){T}_{k}={T}_{k}\left({D}_{k},w)such that for all t≥Tkt\ge {T}_{k}(5.1)‖v(τ,τ−t,θ−τw,vτ−t)‖Vk2+‖∇m+ku(τ,τ−t,θ−τw,uτ−t)‖Vm+k2≤r1k(τ,w).\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{k}}^{2}+\Vert {\nabla }^{m+k}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{u}_{\tau -t}){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w).Let (5.2)Bk(τ,w)={(u,v)∈Vm+k×Vk:‖v‖Vk2+‖u‖Vm+k2≤r1k(τ,w)}.{B}_{k}\left(\tau ,w)=\left\{\left(u,v)\in {V}_{m+k}\times {V}_{k}:\Vert v{\Vert }_{{V}_{k}}^{2}+\Vert u{\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w)\right\}.Then, by (5.2), Bk={Bk(τ,w)}w∈Ω1{B}_{k}={\left\{{B}_{k}\left(\tau ,w)\right\}}_{w\in {\Omega }_{1}}are the closed absorption sets of Φk{\Phi }_{k}in Xk{X}_{k}. We are now ready to prove the asymptotic compactness of Φk{\Phi }_{k}in Xk{X}_{k}.Lemma 5.1Suppose MMsatisfies (M),h(x)∈Vm+k(Ω)\left(M),h\left(x)\in {V}_{m+k}\left(\Omega ), (3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies, (F1)\left({F}_{1}), then Φk{\Phi }_{k}is asymptotically compact in Xk{X}_{k}, that is, for every τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}, the sequence {Φk(ti,τ−ti,θ−tiw,(uτ,i,vτ,i))}\left\{{\Phi }_{k}\left({t}_{i},\tau -{t}_{i},{\theta }_{-{t}_{i}}w,\left({u}_{\tau ,i},{v}_{\tau ,i}))\right\}has a convergent subsequence in Xk{X}_{k}provided ti→∞{t}_{i}\to \infty and(uτ,i,vτ,i)∈Dk(τ−ti,θ−tiw);Dk={Dk(τ,w):τ∈R,w∈Ω1}∈Dk.\left({u}_{\tau ,i},{v}_{\tau ,i})\in {D}_{k}\left(\tau -{t}_{i},{\theta }_{-{t}_{i}}w);\hspace{0.33em}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}.ProofWe first let ti→∞{t}_{i}\to \infty , it follows from Lemma 4.1 that there exist i1=i1(τ,w,Dk)>0{i}_{1}={i}_{1}\left(\tau ,w,{D}_{k})\gt 0such that for every i≥i1i\ge {i}_{1}(5.3)‖v(τ,τ−ti,θ−τw,vτ)‖Vk2+‖u(τ,τ−ti,θ−τw,uτ)‖Vm+k2≤r1k(τ,w),\Vert v\left(\tau ,\tau -{t}_{i},{\theta }_{-\tau }w,{v}_{\tau }){\Vert }_{{V}_{k}}^{2}+\Vert u\left(\tau ,\tau -{t}_{i},{\theta }_{-\tau }w,{u}_{\tau }){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w),next by using Lemma 4.2 for ∀ηk>0\forall \hspace{-0.25em}{\eta }_{k}\gt 0, there are ik2=ik2(ηk,w,Bk){i}_{k2}={i}_{k2}\left({\eta }_{k},w,{B}_{k})and Nk=Nk(ηk,w)>0{N}_{k}={N}_{k}\left({\eta }_{k},w)\gt 0such that for every ik≥ik2{i}_{k}\ge {i}_{k2}(5.4)‖(I−Pn)v(τ,τ−tki,θ−τw,vτ)‖Vk2+‖(I−Pn)u(τ,τ−tki,θ−τw,uτ)‖Vm+k2≤ηk.\Vert \left(I-{P}_{n})v\left(\tau ,\tau -{t}_{ki},{\theta }_{-\tau }w,{v}_{\tau }){\Vert }_{{V}_{k}}^{2}+\Vert \left(I-{P}_{n})u\left(\tau ,\tau -{t}_{ki},{\theta }_{-\tau }w,{u}_{\tau }){\Vert }_{{V}_{m+k}}^{2}\le {\eta }_{k}.By using (5.3), we find that {PNk(u(τ,τ−ti,w),v(τ,τ−ti,w))}\left\{{P}_{{N}_{k}}\left(u\left(\tau ,\tau -{t}_{i},w),v\left(\tau ,\tau -{t}_{i},w))\right\}is bounded in PNkXk{P}_{{N}_{k}}{X}_{k}and PNkXk{P}_{{N}_{k}}{X}_{k}is finite dimensional, which associates with (5.4) implies that {(u(τ,τ−ti,w),v(τ,τ−ti,w))}\left\{\left(u\left(\tau ,\tau -{t}_{i},w),v\left(\tau ,\tau -{t}_{i},w))\right\}is precompact in Xk{X}_{k}.□Theorem 5.2Suppose MMsatisfies (M),h(x)∈Vm+k(Ω)\left(M),\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ), (3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1}), then the family of cocycles Φk{\Phi }_{k}generated by (3.8) has a family of pullback Dk{{\mathcal{D}}}_{k}attractors {Ak}={{Ak(τ,w)}∈Dk(k=1,2,…,m)}\left\{{A}_{k}\right\}=\left\{\left\{{A}_{k}\left(\tau ,w)\right\}\in {{\mathcal{D}}}_{k}\hspace{0.25em}\left(k=1,2,\ldots ,m)\right\}in Xk{X}_{k}and can be expressed as follows: Ak(τ,w)=⋂τ≥0⋃t≥τΦk(t,τ−t,θ−tw,Bk(τ−t,θ−tw))¯,τ∈R,w∈Ω1.{A}_{k}\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }{\Phi }_{k}\left(t,\tau -t,{\theta }_{-t}w,{B}_{k}\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.ProofFrom (5.2), Lemmas 5.1 and 2.10, the conclusion of Theorem 5.2 can be obtained.Theorem 5.2 is proved.□Note 5.3Theorem 5.2 shows the family of cocycles Φk{\Phi }_{k}generated by (3.8) has a unique pullback attractor Ak{A}_{k}, respectively, in the space Xk(k=0,1,…,m){X}_{k}\left(k=0,1,\ldots ,m), which together form a family of pullback attractors {Ak}\left\{{A}_{k}\right\}. At the same time, according to Lemma 4.1 and (5.2) and the tight embedding of Xk↪X0,k=1,2,…,m{X}_{k}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{X}_{0},k=1,2,\ldots ,m, get the corresponding a family of pullback attractors {Ak}\left\{{A}_{k}\right\}, which is (Xk,X0)\left({X}_{k},{X}_{0})the family of random weak attractors, which means that the family of cocycles Φk{\Phi }_{k}has uniformly asymptotically compact absorption sets Bk(τ,w)⊂X0,k=1,2,…,m{B}_{k}\left(\tau ,w)\subset {X}_{0},k=1,2,\ldots ,m, where Bk(τ,w){B}_{k}\left(\tau ,w)are the bounded sets in Xk{X}_{k}, i.e., Φk{\Phi }_{k}are asymptotically compact in X0{X}_{0}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Open Mathematics , Volume 20 (1): 21 – Jan 1, 2022

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de Gruyter
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© 2022 Penghui Lv et al., published by De Gruyter
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2391-5455
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10.1515/math-2022-0003
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Abstract

1IntroductionLet Ω⊂RN\Omega \subset {R}^{N}be a bounded domain with smooth boundary (i.e., the derivative of the function at the boundary exists and is continuous). In this paper, we study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω\Omega : (1.1)utt+a(x)(−Δ)mut+b(x)M(‖∇mu‖)(−Δ)mu+g(x,u)=f(x,t)+h(x)∂w∂t,u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ,u(x,τ)=uτ(x),ut(x,τ)=u1τ(x),x∈Ω,\left\{\begin{array}{l}{u}_{tt}+a\left(x){\left(-\Delta )}^{m}{u}_{t}+b\left(x)M\left(\Vert {\nabla }^{m}u\Vert ){\left(-\Delta )}^{m}u+g\left(x,u)=f\left(x,t)+h\left(x)\frac{\partial w}{\partial t},\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{1em}i=1,2,\ldots ,m-1,\hspace{1em}x\in \Gamma ,\hspace{1em}t\ge \tau ,\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}{u}_{t}\left(x,\tau )={u}_{1\tau }\left(x),\hspace{1em}x\in \Omega ,\end{array}\right.where Γ\Gamma is the smooth boundary of Ω\Omega , vvis the outer normal vector on the boundary Γ\Gamma , m>1m\gt 1, a(x)andb(x)a\left(x)\hspace{0.25em}\text{and}\hspace{0.25em}b\left(x)are variable coefficient functions, f(x,t)∈Lloc2(R,Vk(Ω))f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega ))is a time-dependent external force term, wwis a one-dimensional bilateral standard Wiener process, h(x)∂w∂th\left(x)\frac{\partial w}{\partial t}describes white noise, and g(x,u)g\left(x,u)is a nonlinear function that satisfies certain growth conditions and dissipation conditions.The Kirchhoff model was proposed in 1883 to describe the motion of elastic cross-section. Compared with classical wave equations, the Kirchhoff model can describe the motion of elastic rod more accurately. There has been a lot of in-depth research on the Kirchhoff equations. [1,2,3, 4,5,6] studied the long-term dynamics of the autonomous low-order Kirchhoff equation; [7,8,9, 10,11] studied the existence of global solutions and the blow-up of solutions of the higher-order Kirchhoff equations.Stochastic wave equations are a very important class of stochastic partial differential equations, which are widely used in many fields such as fluid mechanics, physics, electricity, etc. The random attractor is an important tool for studying the long-term asymptotic behavior of stochastic dynamical systems. Using it to characterize the long-term behavior of random dynamical systems has laid a solid foundation for the study of random dynamical systems. After more than 30 years of development, random dynamical systems have also been extensively studied. Many scholars have conducted in-depth studies on the dynamical behavior of random wave equations in unbounded domains [12,13,14, 15,16,17] and bounded domains [18,19,20]. For new trends in functional analysis and random attractors, see also [21,22, 23,24].Regarding the variable coefficients in the equation, it represents the wave velocity at the space coordinate xx, which will appear in the wave phenomena in mathematical physics, marine acoustics, and other fields. It is of great practical significance to study the mathematical and physical equations with variable coefficients. In [25], they studied the global well-posedness and asymptotic behavior of solutions of Kirchhoff-type equations with variable coefficients and weak damping in unbounded domains. More relevant results can also be found in [26,27,28, 29,30,31, 32,33].In recent years, Lin and Chen [34], Lin and Jin [35] have performed a detailed study on the long-term dynamical behavior of higher-order wave equations and proposed the concept of the family of attractors. Combined with the current research results, there are no relevant research results on the long-time dynamics of the nonautonomous stochastic higher-order Kirchhoff equation, and the asymptotic behavior of the higher-order Kirchhoff equation with variable coefficients has not been studied. By studying the nonautonomous stochastic higher-order Kirchhoff model with variable coefficients, the relevant results of the Kirchhoff model can be generalized, and the theoretical achievements of the Kirchhoff model can be enriched, which lays a theoretical foundation for later application. Therefore, this article will specifically study the family of random attractors of nonautonomous random higher-order Kirchhoff equation with variable coefficients. In the research process, the reasonable assumption and Leibniz formula are used to overcome how to define the Lp{L}^{p}-weighted space and the difficulty of estimating the absorption sets and asymptotic compactness caused by the variable coefficients.Section 2 of this article introduces related theories, related definitions, and theories of stochastic dynamical systems; Section 3 presents the family of the continuous cocycle of the problem; In Section 4, the uniform estimation of the solution of problem (1.1) is obtained, and the asymptotic compactness of Φk{\Phi }_{k}is obtained through the decomposition method; in Section 5, we get the family of Dk{{\mathcal{D}}}_{k}-random attractors of Φk{\Phi }_{k}in Xk{X}_{k}.2Preparatory knowledgeIn this section, we mainly give the related theories of nonautonomous stochastic dynamical systems and random attractor (the family of random attractors).First, the relevant notation needed in this paper is introduced: Define the inner product and norm on H=L2(Ω)H={L}^{2}\left(\Omega )as (⋅,⋅)\left(\cdot ,\cdot )and (‖⋅‖)\left(\Vert \cdot \Vert ), Lp=Lp(Ω),‖⋅‖p=‖⋅‖Lp{L}^{p}={L}^{p}\left(\Omega ),\Vert \cdot {\Vert }_{p}=\Vert \cdot {\Vert }_{{L}^{p}}, where p≥1p\ge 1. Set variable coefficient a(x),b(x)=b0a(x),b0a\left(x),b\left(x)={b}_{0}a\left(x),{b}_{0}as a positive constant, satisfying a∈C0∞(Ω),a(x)≥a00>0,∂ia∂iv∣Γ=0,a0=‖a(x)‖∞,a(x)−1=μ(x),x∈Ωa\in {C}_{0}^{\infty }\left(\Omega ),a\left(x)\ge {a}_{00}\gt 0,\frac{{\partial }^{i}a}{{\partial }^{i}v}{| }_{\Gamma }=0,{a}_{0}=\Vert a\left(x){\Vert }_{\infty },a{\left(x)}^{-1}=\mu \left(x),x\in \Omega , and μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ).By D1,2{D}^{1,2}, we define the closure of the C0∞(Ω){C}_{0}^{\infty }\left(\Omega )functions with respect to the “energy norm” ‖u‖D1,2=∫Ω∣∇u∣2dx\Vert u{\Vert }_{{D}^{1,2}}={\int }_{\Omega }| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x. It is well known that D1,2≡D1,2(Ω)={u∈L2N/(N−2)(Ω)∣∇u∈(L2(Ω))N},{D}^{1,2}\equiv {D}^{1,2}\left(\Omega )=\left\{u\in {L}^{2N\text{/}\left(N-2)}\left(\Omega )| \nabla u\in {\left({L}^{2}\left(\Omega ))}^{N}\right\},and for D1,2↪L2N/(N−2)(Ω){D}^{1,2}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{2N\text{/}\left(N-2)}\left(\Omega ), there exists β>0\beta \gt 0such that ‖u‖2N/(N−2)≤β‖u‖D1,2\Vert u{\Vert }_{2N\text{/}\left(N-2)}\le \beta \Vert u{\Vert }_{{D}^{1,2}}.Lemma 2.1[26] Suppose that μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ), then for all u∈C0∞(Ω)u\in {C}_{0}^{\infty }\left(\Omega ), there exists α>0\alpha \gt 0such thatα∫Ωμu2dx≤∫Ω∣∇u∣2dx,\alpha \mathop{\int }\limits_{\Omega }\mu {u}^{2}{\rm{d}}x\le \mathop{\int }\limits_{\Omega }| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x,where α=β−2‖μ‖N/2−1\alpha ={\beta }^{-2}\Vert \mu {\Vert }_{N\hspace{-0.08em}\text{/}\hspace{-0.08em}2}^{-1}.Let μ>0\mu \gt 0be the weight function, and the weighted space Lμp=Lμp(Ω){L}_{\mu }^{p}={L}_{\mu }^{p}\left(\Omega )with the following norm: ‖u‖Lμpp=∫Ωμ∣u∣pdx=‖μ1pu‖pp,\Vert u{\Vert }_{{L}_{\mu }^{p}}^{p}=\mathop{\int }\limits_{\Omega }\mu | u\hspace{-0.25em}{| }^{p}{\rm{d}}x=\Vert {\mu }^{\tfrac{1}{p}}u{\Vert }_{p}^{p},for 1≤p<+∞1\le p\lt +\infty . Clearly Lμ2=Lμ2(Ω){L}_{\mu }^{2}={L}_{\mu }^{2}\left(\Omega )is a separable Hilbert space the inner product and norm are respectively:(u,v)μ=∫Ωμuvdx=μ12u,μ12v,‖u‖Lμ2=‖μ12u‖.{\left(u,v)}_{\mu }=\mathop{\int }\limits_{\Omega }\mu uv{\rm{d}}x=\left({\mu }^{\tfrac{1}{2}}u,{\mu }^{\tfrac{1}{2}}v\right),\hspace{1em}\Vert u{\Vert }_{{L}_{\mu }^{2}}=\Vert {\mu }^{\tfrac{1}{2}}u\Vert .For p:1≤p<∞p:1\le p\lt \infty , the Banach space Lμp{L}_{\mu }^{p}is uniformly convex, reflexive space, and (Lμp)′=Lμp′{\left({L}_{\mu }^{p})}^{^{\prime} }={L}_{\mu }^{p^{\prime} }, where p′{p}^{^{\prime} }is the conjugate number of pp.Lemma 2.2[26] Suppose that μ∈LN2(Ω)∩C0∞(Ω)\mu \in {L}^{\tfrac{N}{2}}\left(\Omega )\cap {C}_{0}^{\infty }\left(\Omega ), then D1,2{D}^{1,2}is compactly embedded in Lμ2{L}_{\mu }^{2}. LetVm=H0m(Ω)=Hm(Ω)∩H01(Ω),Vm+k=H0m+k(Ω)=Hm+k(Ω)∩H01(Ω),k=0,1,…,m,{V}_{m}={H}_{0}^{m}\left(\Omega )={H}^{m}\left(\Omega )\cap {H}_{0}^{1}\left(\Omega ),\hspace{1em}{V}_{m+k}={H}_{0}^{m+k}\left(\Omega )={H}^{m+k}\left(\Omega )\cap {H}_{0}^{1}\left(\Omega ),\hspace{1em}k=0,1,\ldots ,m,and the corresponding inner product and norm are, respectively,(u,v)Vm+k=(∇m+ku,∇m+kv),‖u‖Vm+k=‖∇m+ku‖H.{\left(u,v)}_{{V}_{m+k}}=\left({\nabla }^{m+k}u,{\nabla }^{m+k}v),\hspace{1em}\Vert u{\Vert }_{{V}_{m+k}}=\Vert {\nabla }^{m+k}u{\Vert }_{H}.At the same time, a general form of Poincare inequality: λ1‖∇ru‖2≤‖∇r+1u‖2{\lambda }_{1}\Vert {\nabla }^{r}u{\Vert }^{2}\le \Vert {\nabla }^{r+1}u{\Vert }^{2}, where λ1{\lambda }_{1}is the first eigenvalue of −Δ-\Delta . In the text, Ci{C}_{i}is a positive constant, C(⋅)C\left(\cdot )represents a positive constant that depends on the parameters in parentheses, and Cmn{C}_{m}^{n}is the corresponding number of combinations.Assuming that (X,‖⋅‖X)\left(X,\Vert \cdot {\Vert }_{X})is a separable Hilbert space, and B(X)B\left(X)is the Borel σ\sigma -algebra of X(Ω1,ℱ,P)X\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P)is the metric probability space.Definition 2.3[12] Let θt:R×Ω1→Ω1{\theta }_{t}:R\times {\Omega }_{1}\to {\Omega }_{1}be a family of (B(X)×ℱ,ℱ)\left(B\left(X)\times {\mathcal{ {\mathcal F} }},{\mathcal{ {\mathcal F} }})-measurable mappings such that θ0(⋅){\theta }_{0}\left(\cdot )is the identity on Ω1∀t,s∈R,θt+s(⋅)=θt(⋅)∘θs(⋅),Pθt(⋅)=P{\Omega }_{1}\hspace{0.33em}\forall t,s\in R,{\theta }_{t+s}\left(\cdot )={\theta }_{t}\left(\cdot )\circ {\theta }_{s}\left(\cdot ),P{\theta }_{t}\left(\cdot )=P. A mapping Φ:R+×R×Ω1×X→X\Phi :{R}^{+}\times R\times {\Omega }_{1}\times X\to Xis called a continuous cocycle or continuous random dynamical system (RDS) on XXover RRand (Ω1,ℱ,P,(θt)t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left({\theta }_{t})}_{t\in R})if for all τ∈R,w∈Ω1,t,s∈R+\tau \in R,w\in {\Omega }_{1},t,s\in {R}^{+}the following conditions are satisfied: i:Φ(⋅,τ,⋅,⋅):R+×Ω1×X→X\Phi \left(\cdot ,\hspace{0.25em}\tau ,\cdot ,\cdot ):{R}^{+}\times {\Omega }_{1}\times X\to Xis a (B(R+)×ℱ×B(X),B(X))\left(B\left({R}^{+})\times {\mathcal{ {\mathcal F} }}\times B\left(X),B\left(X))-measurable mapping;ii:Φ(0,τ,w,⋅)\Phi \left(0,\tau ,w,\cdot )is the identity on XX;iii:Φ(t+s,τ,w,⋅)=Φ(t,τ+s,θsw,Φ(s,τ,w,⋅))\Phi \left(t+s,\tau ,w,\cdot )=\Phi \left(t,\tau +s,{\theta }_{s}w,\Phi \left(s,\tau ,w,\cdot ));iv:Φ(t,τ,w,⋅):X→X\Phi \left(t,\tau ,w,\cdot ):X\to Xis continuous.Let D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}be a family of subsets parameterized by (τ,w)∈R×Ω1\left(\tau ,w)\in R\times {\Omega }_{1}in XX.Definition 2.4[13] The family D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}satisfies: (1)for all (τ,w)∈R×Ω1D(τ,w)\left(\tau ,w)\in R\times {\Omega }_{1}\hspace{0.33em}D\left(\tau ,w)is a closed nonempty subset of XX;(2)for every fixed x∈Xx\in Xand any τ∈R\tau \in R, the mapping w∈Ω1→distX(x,B(τ,w))w\in {\Omega }_{1}\to {{\rm{dist}}}_{X}\left(x,B\left(\tau ,w))is (ℱ,B(R+))\left({\mathcal{ {\mathcal F} }},B\left({R}^{+}))measurable, then the family DDis measurable with to ℱ{\mathcal{ {\mathcal F} }}in Ω1{\Omega }_{1}.Definition 2.5[15] For all σ>0,w∈Ω1D={D(τ,w)⊆X:τ∈R,w∈Ω1}\sigma \gt 0,w\in {\Omega }_{1}\hspace{0.33em}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}satisfies: limt→−∞eσt‖D(τ+t,θtw)‖X=0,\mathop{\mathrm{lim}}\limits_{t\to -\infty }{e}^{\sigma t}\Vert D\left(\tau +t,{\theta }_{t}w){\Vert }_{X}=0,then D={D(τ,w)⊆X:τ∈R,w∈Ω1}D=\left\{D\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}is called tempered.Let D=D(X){\mathcal{D}}={\mathcal{D}}\left(X)be the set of all random tempered sets in XX.Definition 2.6[12] A family K={K(τ,w)⊆X:τ∈R,w∈Ω1}∈DK=\left\{K\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}\in {\mathcal{D}}of nonempty subsets of XXis called a measurable D{\mathcal{D}}-pullback attracting(or absorbing) set for {Φ(t,τ,w)}t≥0,τ∈R,w∈Ω1{\left\{\Phi \left(t,\tau ,w)\right\}}_{t\ge 0,\tau \in R,w\in {\Omega }_{1}}if (1)KKis measurable with respect to the PPcompletion of ℱ{\mathcal{ {\mathcal F} }}in Ω1{\Omega }_{1};(2)for all τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}and for every D∈DD\in {\mathcal{D}}, there exists T(D,τ,w)>0T\left(D,\tau ,w)\gt 0such that Φ(t,τ−t,θ−tw,D(τ−t,θ−tw))⊆K(τ,w),∀t≥T(D,τ,w).\Phi \left(t,\tau -t,{\theta }_{-t}w,D\left(\tau -t,{\theta }_{-t}w))\subseteq K\left(\tau ,w),\hspace{1em}\forall t\ge T\left(D,\tau ,w).Definition 2.7[15] Φ\Phi is said to be asymptotically compact in XXif for τ∈R,w∈Ω1,D={D(τ,w)⊆X:τ∈R,w∈Ω1}∈D,xn∈B(τ−tn,θ−tnw){Φ(tn,τ−tn,θ−tnw,xn)}n=1∞\tau \in R,w\in {\Omega }_{1},D=\{D\left(\tau ,w)\hspace{0.25em}\subseteq X:\tau \in R,w\in {\Omega }_{1}\}\in {\mathcal{D}},{x}_{n}\in B\left(\tau -{t}_{n},{\theta }_{-{t}_{n}}w)\hspace{0.33em}{\left\{\Phi \left({t}_{n},\tau -{t}_{n},{\theta }_{-{t}_{n}}w,{x}_{n})\right\}}_{n=1}^{\infty }has a convergent subsequence in XXwhenever tn→∞{t}_{n}\to \infty .Definition 2.8[13] A family A={A(τ,w)⊆X:τ∈R,w∈Ω1}∈DA=\left\{A\left(\tau ,w)\subseteq X:\tau \in R,w\in {\Omega }_{1}\right\}\in {\mathcal{D}}is called a D{\mathcal{D}}-pullback random attractor for {Φ(t,τ,w)}t≥0,τ∈R,w∈Ω1{\left\{\Phi \left(t,\tau ,w)\right\}}_{t\ge 0,\tau \in R,w\in {\Omega }_{1}}if (1)A(τ,w)A\left(\tau ,w)is measurable in Ω1{\Omega }_{1}with respect to ℱ{\mathcal{ {\mathcal F} }}and compact in XXfor ∀τ∈R,w∈Ω1\forall \hspace{-0.25em}\tau \in R,w\in {\Omega }_{1},(2)AAis invariant, i.e., for ∀τ∈R\forall \hspace{-0.25em}\tau \in Rand w∈Ω1,∀t≥0w\in {\Omega }_{1},\forall t\ge 0, Φ(t,τ,w,A(τ,w))=A(t+τ,θtw);\Phi \left(t,\tau ,w,A\left(\tau ,w))=A\left(t+\tau ,{\theta }_{t}w);(3)AAattracts every member of D{\mathcal{D}}, i.e., for every D∈D,τ∈RD\in {\mathcal{D}},\tau \in Rand for every w∈Ω1w\in {\Omega }_{1}, limt→+∞distX(Φ(t,τ−t,θ−tw,B(τ−t,θ−tw)),A(τ,w))=0,\mathop{\mathrm{lim}}\limits_{t\to +\infty }{{\rm{dist}}}_{X}\left(\Phi \left(t,\tau -t,{\theta }_{-t}w,B\left(\tau -t,{\theta }_{-t}w)),A\left(\tau ,w))=0,where distX(P,Q){{\rm{dist}}}_{X}\left(P,Q)denotes the Hausdorff semi-distance between two subsets PPand QQof XX.If we change D=D(X){\mathcal{D}}={\mathcal{D}}\left(X)to Dk=Dk(Xk){{\mathcal{D}}}_{k}={{\mathcal{D}}}_{k}\left({X}_{k}), where k=0,1,…,mk=0,1,\ldots ,m, then AAin Definition 2.8 can be a family of random attractors {Ak}\left\{{A}_{k}\right\}.Lemma 2.9[12] Let D{\mathcal{D}}be a neighborhood-closed collection of (τ,w)\left(\tau ,w)-parametrized families of nonempty subsets of XXand Φ\Phi be a continuous cocycle on XXover RRand (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R}), then Φ\Phi has a pullback D{\mathcal{D}}-attract AAif and only if Φ\Phi is pullback D{\mathcal{D}}asymptotically compact in XXand Φ\Phi has a closed, ℱ{\mathcal{ {\mathcal F} }}-measurable pullback D{\mathcal{D}}-absorbing set KKin D{\mathcal{D}}and the unique pullback D{\mathcal{D}}-attractor A={A(τ,w)}A=\left\{A\left(\tau ,w)\right\}is given byA(τ,w)=⋂τ≥0⋃t≥τΦ(t,τ−t,θ−tw,K(τ−t,θ−tw))¯,τ∈R,w∈Ω1.A\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }\Phi \left(t,\tau -t,{\theta }_{-t}w,K\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.Similarly, Lemma 2.9 can be extended to Lemma 2.10 of the family of pullback attractors.Lemma 2.10Let Dk{{\mathcal{D}}}_{k}be neighborhood-closed collections of (τ,w)\left(\tau ,w)-parametrized families of nonempty subsets of Xk,k=1,2,…,m{X}_{k},k=1,2,\ldots ,m, and Φk{\Phi }_{k}be the family of continuous cocycles on Xk,k=1,2,…,m{X}_{k},k=1,2,\ldots ,mover RRand (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R}), then Φk{\Phi }_{k}has the family of pullback Dk{{\mathcal{D}}}_{k}-attracts {Ak}\left\{{A}_{k}\right\}if and only if Φk{\Phi }_{k}is pullback Dk{{\mathcal{D}}}_{k}-asymptotically compact in Xk{X}_{k}and Dk{{\mathcal{D}}}_{k}has closed, ℱ{\mathcal{ {\mathcal F} }}-measurable pullback Dk{{\mathcal{D}}}_{k}-absorbing sets Kk{K}_{k}in Dk{{\mathcal{D}}}_{k}and the unique pullback Dk{{\mathcal{D}}}_{k}-attractor Ak={Ak(τ,w)}{A}_{k}=\left\{{A}_{k}\left(\tau ,w)\right\}is given byAk(τ,w)=⋂τ≥0⋃t≥τΦk(t,τ−t,θ−tw,Kk(τ−t,θ−tw))¯,τ∈R,w∈Ω1.{A}_{k}\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }{\Phi }_{k}\left(t,\tau -t,{\theta }_{-t}w,{K}_{k}\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.3The family of cocycles of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficientsLet (Ω1,ℱ,P)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P)be a probability space, where Ω1={w∈C(R,R),w(0)=0}.{\Omega }_{1}=\left\{w\in C\left(R,R),w\left(0)=0\right\}.wwis a two-sided real-valued Winner processes on the probability space (Ω1,ℱ,P)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P). Define θtw(⋅)=w(⋅+t)−w(t),w∈Ω1,t∈R{\theta }_{t}w\left(\cdot )=w\left(\cdot +t)-w\left(t),w\in {\Omega }_{1},t\in R, thus, (Ω1,ℱ,P,(θt)t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left({\theta }_{t})}_{t\in R})is an ergodic metric dynamical system.For a small positive number ε\varepsilon , let zzbe a new variable given by z=ut+εuz={u}_{t}+\varepsilon uand then, system (1.1) becomes (3.1)∂u∂t+εu=z;∂z∂t=εz−a(x)(−Δ)mz+εa(x)(−Δ)mu−ε2u−b(x)M(‖∇mu‖2)(−Δ)mu−g(x,u)+f(x,t)+h(x)dwdt;u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ;u(x,τ)=uτ(x),z(x,τ)=zτ(x)=u1τ(x)+uτ(x),x∈Ω,\left\{\begin{array}{l}\frac{\partial u}{\partial t}+\varepsilon u=z;\\ \frac{\partial z}{\partial t}=\varepsilon z-a\left(x){\left(-\Delta )}^{m}z+\varepsilon a\left(x){\left(-\Delta )}^{m}u-{\varepsilon }^{2}u-b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u-g\left(x,u)+f\left(x,t)+h\left(x)\frac{{\rm{d}}w}{{\rm{d}}t};\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{0.33em}i=1,2,\ldots ,m-1,\hspace{0.33em}x\in \Gamma ,\hspace{0.33em}t\ge \tau ;\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}z\left(x,\tau )={z}_{\tau }\left(x)={u}_{1\tau }\left(x)+{u}_{\tau }\left(x),\hspace{0.33em}x\in \Omega ,\end{array}\right.where (M)M∈C1(R+),M′≥0\left(M)M\in {C}^{1}\left({R}^{+}),M^{\prime} \ge 0, and M(s)≤M0⋅(1+sq),0<q<1/2,M0=M(0)M\left(s)\le {M}_{0}\cdot \left(1+{s}^{q}),0\lt q\lt 1\hspace{-0.08em}\text{/}\hspace{-0.08em}2,{M}_{0}=M\left(0)is a positive constant ∀s∈R+\forall \hspace{-0.25em}s\in {R}^{+}, h(x)∈Vm+k(Ω),x∈Ω,t≥τ,τ∈R,k=0,1,…,m,f(x,t)∈Lloc2(R,Vk(Ω))h\left(x)\in {V}_{m+k}\left(\Omega ),x\in \Omega ,t\ge \tau ,\tau \in R,k=0,1,\ldots ,m,f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega )). In order to get the conclusion of this article, suppose that the nonlinear term g(x,u)g\left(x,u)satisfies the following conditions: for ∀u∈R,x∈Ω\forall \hspace{-0.25em}u\in R,x\in \Omega , there are positive constants c1,c2,c3,c4,c5>0{c}_{1},{c}_{2},{c}_{3},{c}_{4},{c}_{5}\gt 0, satisfying (3.2)∣g(x,u)∣≤c1∣p∣p+ϕ1(x),ϕ1∈Lμ2(Ω),\hspace{-17.6em}| g\left(x,u)| \le {c}_{1}| p\hspace{-0.25em}{| }^{p}+{\phi }_{1}\left(x),\hspace{0.33em}{\phi }_{1}\in {L}_{\mu }^{2}\left(\Omega ),(3.3)ug(x,u)−c2G(x,u)≥ϕ2(x),ϕ2∈Lμ1(Ω),ug\left(x,u)-{c}_{2}G\left(x,u)\ge {\phi }_{2}\left(x),\hspace{0.33em}{\phi }_{2}\in {L}_{\mu }^{1}\left(\Omega ),(3.4)G(x,u)≥c3∣u∣p+1−ϕ3(x),ϕ3∈Lμ1(Ω),\hspace{-17.6em}G\left(x,u)\ge {c}_{3}| u\hspace{-0.25em}{| }^{p+1}-{\phi }_{3}\left(x),\hspace{0.33em}{\phi }_{3}\in {L}_{\mu }^{1}\left(\Omega ),(3.5)∣gu(x,u)∣≤c4∣u∣p−1+ϕ4(x),ϕ4∈Vm(Ω),\hspace{-17.6em}| {g}_{u}\left(x,u)| \le {c}_{4}| u\hspace{-0.25em}{| }^{p-1}+{\phi }_{4}\left(x),\hspace{0.33em}{\phi }_{4}\in {V}_{m}\left(\Omega ),(3.6)∣∇xkg(x,u)∣≤c5∣u∣p+ϕ5(x),ϕ5∈Vk(Ω),\hspace{-17.6em}| {\nabla }_{x}^{k}g\left(x,u)| \le {c}_{5}| u\hspace{-0.25em}{| }^{p}+{\phi }_{5}\left(x),\hspace{0.33em}{\phi }_{5}\in {V}_{k}\left(\Omega ),where 1≤p<+∞1\le p\lt +\infty , for N=1,2;N=1,2;1≤p<NN−21\le p\lt \frac{N}{N-2}N=3,4N=3,4; and G(x,u)=∫0ug(x,s)dsG\left(x,u)={\int }_{0}^{u}g\left(x,s){\rm{d}}s. From equations (3.2) and (3.3), we can get (3.7)G(x,u)≤c6(∣u∣2+∣u∣p+1+ϕ12+ϕ2).G\left(x,u)\le {c}_{6}\left(| u\hspace{-0.25em}{| }^{2}+| u\hspace{-0.25em}{| }^{p+1}+{\phi }_{1}^{2}+{\phi }_{2}).To show that problem (3.1) generates a random dynamical system, we let v(t,τ,w)=z(t,τ,w)−hw(t)v\left(t,\tau ,w)=z\left(t,\tau ,w)-hw\left(t), and then, (3.1) can be rewritten as the equivalent system with random coefficients but without white noise: (3.8)∂u∂t−v+εu=hw(t);∂v∂t=εv−a(x)(−Δ)mv+εa(x)(−Δ)mu−ε2u−b(x)M(‖∇mu‖2)(−Δ)mu−g(x,u)+f(x,t)+εh(x)w(t)−a(x)(−Δ)mh(x)w(t);u=0,∂iu∂vi=0,i=1,2,…,m−1,x∈Γ,t≥τ;u(x,τ)=uτ(x),v(x,τ)=vτ(x)=zτ(x)−hw(τ),x∈Ω.\left\{\begin{array}{l}\frac{\partial u}{\partial t}-v+\varepsilon u=hw\left(t);\\ \frac{\partial v}{\partial t}=\varepsilon v-a\left(x){\left(-\Delta )}^{m}v+\varepsilon a\left(x){\left(-\Delta )}^{m}u-{\varepsilon }^{2}u-b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u\\ -g\left(x,u)+f\left(x,t)+\varepsilon h\left(x)w\left(t)-a\left(x){\left(-\Delta )}^{m}h\left(x)w\left(t);\\ u=0,\hspace{1em}\frac{{\partial }^{i}u}{\partial {v}^{i}}=0,\hspace{0.33em}i=1,2,\ldots ,m-1,\hspace{0.33em}x\in \Gamma ,\hspace{0.33em}t\ge \tau ;\\ u\left(x,\tau )={u}_{\tau }\left(x),\hspace{1em}v\left(x,\tau )={v}_{\tau }\left(x)={z}_{\tau }\left(x)-hw\left(\tau ),\hspace{0.33em}x\in \Omega .\end{array}\right.Let Xk=Vm+k×Vk,k=0,1,…,m{X}_{k}={V}_{m+k}\times {V}_{k},k=0,1,\ldots ,m, when k=0V0=Lμ2k=0\hspace{0.33em}{V}_{0}={L}_{\mu }^{2}, endowed with the usual norm ‖(u,v)‖Xk2=‖u‖Vm+k2+‖v‖Vk2\Vert \left(u,v){\Vert }_{{X}_{k}}^{2}=\Vert u{\Vert }_{{V}_{m+k}}^{2}+\Vert v{\Vert }_{{V}_{k}}^{2}. By the standard Galerkin method: If the assumptions (M)h(x)∈Vm+k(Ω),x∈Ω,t≥τ,τ∈R,f(x,t)∈Lloc2(R,Vk(Ω))\left(M)\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ),x\in \Omega ,t\ge \tau ,\tau \in R,f\left(x,t)\in {L}_{{\rm{loc}}}^{2}\left(R,{V}_{k}\left(\Omega ))conditions (3.2)–(3.6) hold the problem (3.8) is well posed in Xk=Vm+k×Vk{X}_{k}={V}_{m+k}\times {V}_{k}, i.e., for all τ∈R\tau \in Rand P−a.e.w∈Ω1,(uτ,vτ)∈XkP-a.e.w\in {\Omega }_{1},\left({u}_{\tau },{v}_{\tau })\in {X}_{k}, the problem (3.8) has a unique global solution (u(t,τ,w,uτ),(u\left(t,\tau ,w,{u}_{\tau }),v(t,τ,w,vτ))∈C([τ,∞),Xk)v\left(t,\tau ,w,{v}_{\tau }))\in C\left({[}\tau ,\infty ),{X}_{k})and (u(τ,τ,w,uτ),v(τ,τ,w,vτ))=(uτ,vτ)\left(u\left(\tau ,\tau ,w,{u}_{\tau }),v\left(\tau ,\tau ,w,{v}_{\tau }))=\left({u}_{\tau },{v}_{\tau }). Moreover, for t≥τ,(u(t,τ,w,uτ),t\ge \tau ,(u\left(t,\tau ,w,{u}_{\tau }),v(t,τ,w,vτ))v\left(t,\tau ,w,{v}_{\tau }))is (ℱ,B(Xk))\left({\mathcal{ {\mathcal F} }},B\left({X}_{k}))measurable in wwand continuous in (uτ,vτ)\left({u}_{\tau },{v}_{\tau })with respect to the Xk{X}_{k}norm. Thus, the solution mapping can be used to define a family of continuous cocycles for (3.8). Let Φk:R+×R×Ω1×{\Phi }_{k}:{R}^{+}\times R\times {\Omega }_{1}\times Xk→Xk{X}_{k}\to {X}_{k}be mappings given by (3.9)Φk(t,τ,w,(uτ,vτ))=(u(t+τ,τ,θ−τw,uτ),v(t+τ,τ,θ−τw,vτ)),{\Phi }_{k}\left(t,\tau ,w,\left({u}_{\tau },{v}_{\tau }))=\left(u\left(t+\tau ,\tau ,{\theta }_{-\tau }w,{u}_{\tau }),v\left(t+\tau ,\tau ,{\theta }_{-\tau }w,{v}_{\tau })),where (t,τ,w,(uτ,vτ))∈R+×R×Ω1×Xk\left(t,\tau ,w,\left({u}_{\tau },{v}_{\tau }))\in {R}^{+}\times R\times {\Omega }_{1}\times {X}_{k}, then Φk{\Phi }_{k}is a family of continuous cocycles over (R,τ+t)\left(R,\tau +t)and (Ω1,ℱ,P,{θt}t∈R)\left({\Omega }_{1},{\mathcal{ {\mathcal F} }},P,{\left\{{\theta }_{t}\right\}}_{t\in R})on Xk{X}_{k}. For P−a.e.w∈Ω1P-a.e.w\in {\Omega }_{1}and t,s≥0,τ∈R:t,s\ge 0,\hspace{0.33em}\tau \in R:(3.10)Φk(t+s,τ,w,(uτ,vτ))=Φk(t,s+τ,w,Φk(s,τ,w,(uτ,vτ))).{\Phi }_{k}\left(t+s,\tau ,w,\left({u}_{\tau },{v}_{\tau }))={\Phi }_{k}\left(t,s+\tau ,w,{\Phi }_{k}\left(s,\tau ,w,\left({u}_{\tau },{v}_{\tau }))).For any bounded nonempty subset Bk{B}_{k}of Xk{X}_{k}denote by ‖Bk‖=supΦk∈R‖Φ‖Xk\Vert {B}_{k}\Vert ={\sup }_{{\Phi }_{k}\in R}\Vert \Phi {\Vert }_{{X}_{k}}. Let Dk={Dk(τ,w):τ∈R,w∈Ω1}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}be a family of bounded nonempty subsets of Xk{X}_{k}, and for all τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}, (3.11)lims→−∞eσs‖Dk(τ+s,θsw)‖Xk2=0.\mathop{\mathrm{lim}}\limits_{s\to -\infty }{e}^{\sigma s}\Vert {D}_{k}\left(\tau +s,{\theta }_{s}w){\Vert }_{{X}_{k}}^{2}=0.Remember that Dk{{\mathcal{D}}}_{k}is the set of the aforementioned subset family Dk{D}_{k}, that is, Dk={Dk={Dk(τ,w):{{\mathcal{D}}}_{k}=\{{D}_{k}=\{{D}_{k}\left(\tau ,w)\hspace{0.25em}:τ∈R,w∈Ω1}:Dksatisfies (3.11)}\tau \in R,w\in {\Omega }_{1}\}:{D}_{k}\hspace{0.33em}\hspace{0.1em}\text{satisfies (3.11)}\hspace{0.1em}\}.4Uniform estimates of solutionsTo prove the existence of the family of random attractors, we conduct uniform estimates on the solutions of the problem (3.8) defined on Ω\Omega , for the purposes of showing the existence of a family of Dk{{\mathcal{D}}}_{k}pullback absorbing sets and the pullback Dk{{\mathcal{D}}}_{k}asymptotic compactness of the random dynamical system. Let ε>0\varepsilon \gt 0be small enough and satisfy αλ1m−1−3ε>0,2a00λ1m−(a00λ1m+12)ε>0,M0−52ε>0,b0M08−ε>0\alpha {\lambda }_{1}^{m-1}-3\varepsilon \gt 0,2{a}_{00}{\lambda }_{1}^{m}-\left({a}_{00}{\lambda }_{1}^{m}+12)\varepsilon \gt 0,{M}_{0}-\frac{5}{2}\varepsilon \gt 0,\frac{{b}_{0}{M}_{0}}{8}-\varepsilon \gt 0, (4.1)σ=12minαλ1m−1−3ε,ε2,εc22,σ1=12min{2a00λ1m−(a00λ1m+12)ε,ε}.\sigma =\frac{1}{2}{\rm{\min }}\left\{\alpha {\lambda }_{1}^{m-1}-3\varepsilon ,\frac{\varepsilon }{2},\frac{\varepsilon {c}_{2}}{2}\right\},\hspace{1em}{\sigma }_{1}=\frac{1}{2}\min \left\{2{a}_{00}{\lambda }_{1}^{m}-\left({a}_{00}{\lambda }_{1}^{m}+12)\varepsilon ,\varepsilon \right\}.To obtain uniform estimates of the solutions, f(x,t)f\left(x,t)needs to satisfy (F1)∫−∞teσs‖f(⋅,s)‖Vk2ds<∞\left({F}_{1}){\int }_{-\infty }^{t}{e}^{\sigma s}\Vert f\left(\cdot ,\hspace{0.33em}s){\Vert }_{{V}_{k}}^{2}{\rm{d}}s\lt \infty .Lemma 4.1Suppose MMsatisfies (M),h(x)∈Vm+k(Ω),k=0,1,…,m\left(M),\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ),k=0,1,\ldots ,m, (3.2)–(3.6) hold, f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1}), and Bk={Bk(τ,w):τ∈R,w∈Ω1}∈Dk{B}_{k}=\left\{{B}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}for P−a.e.w∈Ω1,τ∈RP-a.e.w\in {\Omega }_{1},\tau \in Rinitial value satisfies (uτ−t,vτ−t)∈Bk(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{k}(\tau -t,{\theta }_{-\tau }w), there exists Tk=Tk(τ,w,Bk)>0{T}_{k}={T}_{k}\left(\tau ,w,{B}_{k})\gt 0such that for all t≥Tkt\ge {T}_{k}, the solution (u(τ,τ,w,uτ−t),v(τ,τ,w,vτ−t))=(uτ−t,vτ−t)(u\left(\tau ,\tau ,w,{u}_{\tau -t}),v(\tau ,\tau ,w,{v}_{\tau -t}))=\left({u}_{\tau -t},{v}_{\tau -t})of problem (3.8) satisfies‖v(τ,τ−t,θ−τw,vτ−t)‖Vk2+‖u(τ,τ−t,θ−τw,vτ−t)‖Vm+k2≤r1k(τ,w),\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{k}}^{2}+\Vert u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w),where r1k(τ,w){r}_{1k}\left(\tau ,w)will be given in detail later.ProofTaking the inner product of (3.8) with vvin Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ), we find that (4.2)12ddt‖v‖Lμ22=ε‖v‖Lμ22−‖∇mv‖2+ε((−Δ)mu,v)−ε2(u,v)Lμ2−b0(M(‖∇mu‖2)(−Δ)mu,v)−(g(x,u),v)Lμ2+(f(x,t),v)Lμ2+εw(t)(h,v)Lμ2−w(t)((−Δ)mh,v),\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}& =& \varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}-\Vert {\nabla }^{m}v{\Vert }^{2}+\varepsilon \left({\left(-\Delta )}^{m}u,v)-{\varepsilon }^{2}{\left(u,v)}_{{L}_{\mu }^{2}}-{b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,v)\\ & & -{\left(g\left(x,u),v)}_{{L}_{\mu }^{2}}+{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+\varepsilon w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-w\left(t)\left({\left(-\Delta )}^{m}h,v),\end{array}for each term on the right-hand side of (4.2): (4.3)ε((−Δ)mu,v)=ε((−Δ)mu,ut+εu+hw(t))=ε2ddt‖∇mu‖2+ε2‖∇mu‖2−εw(t)((−Δ)mu,h),\varepsilon \left({\left(-\Delta )}^{m}u,v)=\varepsilon \left({\left(-\Delta )}^{m}u,{u}_{t}+\varepsilon u+hw\left(t))=\frac{\varepsilon }{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}-\varepsilon w\left(t)\left({\left(-\Delta )}^{m}u,h),(4.4)ε2(u,v)Lμ2=ε2(u,ut+εu−hw(t))Lμ2=ε22ddt‖v‖Lμ22+ε3‖v‖Lμ22−ε2w(t)(u,h)Lμ2,\hspace{-35.1em}{\varepsilon }^{2}{\left(u,v)}_{{L}_{\mu }^{2}}={\varepsilon }^{2}{\left(u,{u}_{t}+\varepsilon u-hw\left(t))}_{{L}_{\mu }^{2}}=\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{3}\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}-{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}},(4.5)b0(M(‖∇mu‖2)(−Δ)mu,v)=b0(M(‖∇mu‖2)(−Δ)mu,ut+εu−hw(t))=b02ddt∫0‖∇mu‖2M(s)ds+εb0M(‖∇mu‖2)‖∇mu‖2−b0M(‖∇mu‖2)w(t)((−Δ)mu,h),\hspace{-37.95em}\begin{array}{l}{b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,v)={b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{u}_{t}+\varepsilon u-hw\left(t))\\ \hspace{1.0em}=\frac{{b}_{0}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s)\hspace{0.33em}{\rm{d}}s+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}u{\Vert }^{2}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}u,h),\end{array}(4.6)(g(x,u),v)Lμ2=(g(x,u),ut+εu−hw(t))Lμ2=ddt∫ΩμG(x,u)dx+ε(g(x,u),u)Lμ2−w(t)(g(x,u),h)Lμ2.{\left(g\left(x,u),v)}_{{L}_{\mu }^{2}}={\left(g\left(x,u),{u}_{t}+\varepsilon u-hw\left(t))}_{{L}_{\mu }^{2}}=\frac{{\rm{d}}}{{\rm{d}}t}\mathop{\int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}-w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}.Substitute (4.3)–(4.6) into (4.2) to obtain (4.7)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+2‖∇mv‖2−2ε‖v‖Lμ22+2εb0M(‖∇mu‖2)‖∇mu‖2−2ε2‖∇mu‖2+2ε3‖u‖Lμ22+2ε(g(x,u),u)Lμ2=2(f(x,t),v)Lμ2+(2b0M(‖∇mu‖2)−2ε)w(t)((−Δ)mu,h)−2ε2w(t)(u,h)Lμ2+2w(t)(g(x,u),h)Lμ2+2w(t)(h,v)Lμ2−2w(t)((−Δ)mh,v).\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+2\Vert {\nabla }^{m}v{\Vert }^{2}\\ \hspace{1.0em}-2\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}u{\Vert }^{2}-2{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+2{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=2{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+\left(2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-2\varepsilon )w\left(t)\left({\left(-\Delta )}^{m}u,h)-2{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+2w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}+2w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-2w\left(t)\left({\left(-\Delta )}^{m}h,v).\end{array}Using the Cauchy-Schwarz inequality, Young’s inequality and Holder’s inequality, we have (4.8)2ε2w(t)(u,h)Lμ2≤ε3‖u‖Lμ22+ε∣w(t)∣2‖h‖Lμ22,2{\varepsilon }^{2}w\left(t){\left(u,h)}_{{L}_{\mu }^{2}}\le {\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},(4.9)2b0M(‖∇mu‖2)w(t)((−Δ)mu,h)≤2b0M0(1+‖∇mu‖2q)∣w(t)∣‖∇mu‖‖∇mh‖≤2b0M0∣w(t)∣‖∇mu‖‖∇mh‖+2b0M0∣w(t)∣‖∇mu‖2q+1‖∇mh‖≤εb0M04‖∇mu‖2+4ε−1b0M0∣w(t)∣2‖∇mh‖2+εb0M04‖∇mu‖2+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q.\begin{array}{l}2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}u,h)\\ \hspace{1.0em}\le \hspace{0.33em}2{b}_{0}{M}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})| w\left(t)| \Vert {\nabla }^{m}u\Vert \Vert {\nabla }^{m}h\Vert \\ \hspace{1.0em}\le 2{b}_{0}{M}_{0}| w\left(t)| \Vert {\nabla }^{m}u\Vert \Vert {\nabla }^{m}h\Vert +2{b}_{0}{M}_{0}| w\left(t)| \Vert {\nabla }^{m}u{\Vert }^{2q+1}\Vert {\nabla }^{m}h\Vert \\ \hspace{1.0em}\le \hspace{0.25em}\frac{\varepsilon {b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m}u{\Vert }^{2}+4{\varepsilon }^{-1}{b}_{0}{M}_{0}| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\frac{\varepsilon {b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m}u{\Vert }^{2}\\ \hspace{1.85em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}.\end{array}By (3.2) and (3.4), we get (4.10)2w(t)(g(x,u),h)Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣∫Ωμ∣u∣p+1dxpp+1‖h‖Lμp+1≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣∫Ω(μG(x,u)+μϕ3)dxpp+1‖h‖Lμp+1≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫ΩμG(x,u)dx+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1,\begin{array}{l}2w\left(t){\left(g\left(x,u),h)}_{{L}_{\mu }^{2}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| {\left(\mathop{\displaystyle \int }\limits_{\Omega }\mu | u{| }^{p+1}{\rm{d}}x\right)}^{\tfrac{p}{p+1}}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| {\left(\mathop{\displaystyle \int }\limits_{\Omega }\left(\mu G\left(x,u)+\mu {\phi }_{3}){\rm{d}}x\right)}^{\tfrac{p}{p+1}}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x\hspace{.85em}+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1},\end{array}(4.11)2(f(x,t),v)Lμ2+2w(t)(h,v)Lμ2−2w(t)((−Δ)mh,v)≤2‖f‖Lμ2‖v‖Lμ2+2ε∣w(t)∣‖h‖Lμ2‖v‖Lμ2+2∣w(t)∣‖∇mh‖‖∇mv‖≤2α−12λ1−m−12‖f‖Lμ2‖∇mv‖+ε‖v‖Lμ22+ε∣w(t)∣2‖h‖Lμ22+12‖∇mv‖2+2∣w(t)∣2‖∇mh‖2≤‖∇mv‖2+ε‖v‖Lμ22+2α−1λ11−m‖f‖Lμ22+2∣w(t)∣2‖∇mh‖2+ε∣w(t)∣2‖h‖Lμ22,\hspace{-35.5em}\begin{array}{l}2{(f\left(x,t),v)}_{{L}_{\mu }^{2}}+2w\left(t){\left(h,v)}_{{L}_{\mu }^{2}}-2w\left(t)\left({\left(-\Delta )}^{m}h,v)\\ \hspace{1.0em}\le \hspace{-0.25em}2\Vert f{\Vert }_{{L}_{\mu }^{2}}\Vert v{\Vert }_{{L}_{\mu }^{2}}+2\varepsilon | w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}\Vert v{\Vert }_{{L}_{\mu }^{2}}+2| w\left(t)| \Vert {\nabla }^{m}h\Vert \Vert {\nabla }^{m}v\Vert \\ \hspace{1.0em}\le \hspace{-0.25em}2{\alpha }^{-\tfrac{1}{2}}{\lambda }_{1}^{-\frac{m-1}{2}}\Vert f{\Vert }_{{L}_{\mu }^{2}}\Vert {\nabla }^{m}v\Vert +\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+\frac{1}{2}\Vert {\nabla }^{m}v{\Vert }^{2}+2| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}\\ \hspace{1.0em}\le \Vert {\nabla }^{m}v{\Vert }^{2}+\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+2| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},\end{array}(4.12)2εw(t)((−Δ)mu,h)≤2ε∣w(t)∣‖∇mh‖‖∇mu‖≤ε2‖∇mu‖2+∣w(t)∣2‖∇mh‖2.\hspace{-35.5em}2\varepsilon w\left(t)\left({\left(-\Delta )}^{m}u,h)\le 2\varepsilon | w\left(t)| \Vert {\nabla }^{m}h\Vert \Vert {\nabla }^{m}u\Vert \le {\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}.Substitute (4.8)–(4.12) into (4.7) to obtain (4.13)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+‖∇mv‖2−3ε‖v‖Lμ22+2εM(‖∇mu‖2)−εM02b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+2ε(g(x,u),u)Lμ2\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+\Vert {\nabla }^{m}v{\Vert }^{2}\\ \hspace{1.0em}-3\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\frac{\varepsilon {M}_{0}}{2}\right){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}-3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\end{array}≤2α−1λ11−m‖f‖Lμ22+(3+4ε−1M0b0)∣w(t)∣2‖∇mh‖2+2ε∣w(t)∣2‖h‖Lμ22+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q+2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫ΩμG(x,u)dx+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1.\begin{array}{l}\hspace{1.0em}\le \hspace{-0.25em}2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}+2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\\ \hspace{2.0em}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}.\end{array}By condition (3.3), we have (4.14)2ε(g(x,u),u)Lμ2≥2εc2∫ΩμG(x,u)dx+∫Ωμϕ2(x)dx.2\varepsilon {\left(g\left(x,u),u)}_{{L}_{\mu }^{2}}\ge 2\varepsilon \left({c}_{2}\mathop{\int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+\mathop{\int }\limits_{\Omega }\mu {\phi }_{2}\left(x){\rm{d}}x\right).Substitute (4.14) into (4.12) to obtain (4.15)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+‖∇mv‖2−3ε‖v‖Lμ22+2εM(‖∇mu‖2)−εM02b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+εc2∫ΩμG(x,u)dx+2ε∫Ωμϕ2(x)dx≤2α−1λ11−m‖f‖Lμ22+(3+4ε−1M0b0)∣w(t)∣2‖∇mh‖2+2ε∣w(t)∣2‖h‖Lμ22+1−2q2ε8q+42q+12q−1b0M0∣w(t)∣21−2q‖∇mh‖21−2q+2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+εc2∫Ωμϕ3(x)dx+(2c1)p+1(εc2)−pp+1p+1p−p∣w(t)∣p+1‖h‖Lμp+1p+1.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]+\Vert {\nabla }^{m}v{\Vert }^{2}-3\varepsilon \Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\frac{\varepsilon {M}_{0}}{2}\right){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}-3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x+2\varepsilon \mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{2}\left(x){\rm{d}}x\\ \hspace{1.0em}\le 2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}| w\left(t){| }^{\tfrac{2}{1-2q}}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}}+2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x+{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}| w\left(t){| }^{p+1}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}.\end{array}According to (4.1), we get (4.16)ddt‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx+σ‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx≤2α−1λ11−m‖f‖Lμ22+C01(1+∣w(t)∣2+∣w(t)∣21−2q+∣w(t)∣p+1),\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}+\sigma \left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s)\hspace{0.33em}{\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}\le 2{\alpha }^{-1}{\lambda }_{1}^{1-m}\Vert f{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{01}\left(1+| w\left(t){| }^{2}+| w\left(t){| }^{\tfrac{2}{1-2q}}+| w\left(t){| }^{p+1}),\end{array}where C01=max(3+4ε−1M0b0)‖∇mh‖2+2ε‖h‖Lμ22+‖ϕ1‖Lμ22‖h‖Lμ22,εc2∫Ωμϕ3(x)dx,1−2q2ε8q+42q+12q−1b0M0‖∇mh‖21−2q,(2c1)p+1(εc2)−pp+1p+1p−p‖h‖Lμp+1p+1.\begin{array}{rcl}{C}_{01}& =& \max \left\{\left(3+4{\varepsilon }^{-1}{M}_{0}{b}_{0})\Vert {\nabla }^{m}h{\Vert }^{2}+2\varepsilon \Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2},\varepsilon {c}_{2}\mathop{\displaystyle \int }\limits_{\Omega }\mu {\phi }_{3}\left(x){\rm{d}}x,\right.\\ & & \left.\frac{1-2q}{2}{\left(\frac{\varepsilon }{8q+4}\right)}^{\tfrac{2q+1}{2q-1}}{b}_{0}{M}_{0}\Vert {\nabla }^{m}h{\Vert }^{\tfrac{2}{1-2q}},{\left(2{c}_{1})}^{p+1}\frac{{\left(\varepsilon {c}_{2})}^{-p}}{p+1}{\left(\frac{p+1}{p}\right)}^{-p}\Vert h{\Vert }_{{L}_{\mu }^{p+1}}^{p+1}\right\}.\end{array}Using the Gronwall inequality to integrate (4.16) over [τ−t,τ]\left[\tau -t,\tau ]with t≥0t\ge 0and replacing wwby θ−τw{\theta }_{-\tau }w, we obtain (4.17)eστ‖v‖Lμ22+b0∫0‖∇mu‖2M(s)ds−ε‖∇mu‖2+ε2‖u‖Lμ22+2∫ΩμG(x,u)dx≤eσ(τ−t)‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx+2α−1λ11−m∫τ−tτeσξ‖f(⋅,ξ)‖Lμ22dξ+C01∫τ−tτeσξ(1+∣w(ξ)∣2+∣w(ξ)∣21−2q+∣w(ξ)∣p+1)dξ,\begin{array}{l}{e}^{\sigma \tau }\left[\Vert v{\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u{\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{2}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u){\rm{d}}x\right]\\ \hspace{1.0em}\le {e}^{\sigma \left(\tau -t)}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}\right.\\ \hspace{2.0em}\left.+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]+2{\alpha }^{-1}{\lambda }_{1}^{1-m}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \\ \hspace{1.0em}\hspace{1.0em}+{C}_{01}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\left(1+| w\left(\xi ){| }^{2}+| w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| w\left(\xi ){| }^{p+1}){\rm{d}}\xi ,\end{array}then (4.18)‖v(τ,τ−t,θ−τw,vτ−t)‖Lμ22+b0∫0‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2M(s)ds−ε‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2+ε2‖u(τ,τ−t,θ−τw,vτ−t)‖Lμ22+2∫ΩμG(x,u(τ,τ−t,θ−τw,vτ−t))dx≤e−σt‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx+2α−1λ11−me−στ∫τ−tτeσξ‖f(⋅,ξ)‖Lμ22dξ+C01e−στ∫τ−tτeσξ(1+∣θ−τw(ξ)∣2+∣θ−τw(ξ)∣21−2q+∣θ−τw(ξ)∣p+1)dξ.\begin{array}{l}\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}+{\varepsilon }^{2}\Vert u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t})){\rm{d}}x\\ \hspace{1.0em}\le {e}^{-\sigma t}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}\right.\\ \hspace{2.0em}\left.+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]+2{\alpha }^{-1}{\lambda }_{1}^{1-m}{e}^{-\sigma \tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \\ \hspace{2.0em}+{C}_{01}{e}^{-\sigma \tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\left(1+| {\theta }_{-\tau }w\left(\xi ){| }^{2}+| {\theta }_{-\tau }w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| {\theta }_{-\tau }w\left(\xi ){| }^{p+1}){\rm{d}}\xi .\end{array}By (3.7), we have (4.19)∫ΩμG(x,u(τ−t))dx≤c6(‖ϕ1‖Lμ22+‖ϕ2‖Lμ1+‖u‖Lμ22+‖u‖Lμp+1p+1)≤C02(1+‖u‖Lμ22+‖u‖Vmp+1).\mathop{\int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\le {c}_{6}\left(\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\phi }_{2}{\Vert }_{{L}_{\mu }^{1}}+\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert u{\Vert }_{{L}_{\mu }^{p+1}}^{p+1})\le {C}_{02}\left(1+\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert u{\Vert }_{{V}_{m}}^{p+1}).Since (u(τ−t),v(τ−t))∈B0(τ−t,θ−τw)\left(u\left(\tau -t),v\left(\tau -t))\in {B}_{0}\left(\tau -t,{\theta }_{-\tau }w)when t→+∞t\to +\infty (4.20)e−σt‖v(τ−t)‖Lμ22+b0∫0‖∇mu(τ−t)‖2M(s)ds−ε‖∇mu(τ−t)‖2+ε2‖u(τ−t)‖Lμ22+2∫ΩμG(x,u(τ−t))dx≤C03e−σt(1+‖v(τ−t)‖Lμ22+‖∇mu(τ−t)‖2+‖∇mu(τ−t)‖p+1)→0,\begin{array}{l}{e}^{-\sigma t}\left[\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+{b}_{0}\underset{0}{\overset{\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}}{\displaystyle \int }}M\left(s){\rm{d}}s-\varepsilon \Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert u\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2\mathop{\displaystyle \int }\limits_{\Omega }\mu G\left(x,u\left(\tau -t)){\rm{d}}x\right]\\ \hspace{1.0em}\le {C}_{03}{e}^{-\sigma t}\left(1+\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{p+1})\to 0,\end{array}and there exists T0=T0(τ,w,B0){T}_{0}={T}_{0}\left(\tau ,w,{B}_{0})such that for all t≥T0t\ge {T}_{0}, (4.21)C03e−σt(1+‖v(τ−t)‖Lμ22+‖∇mu(τ−t)‖2+‖∇mu(τ−t)‖p+1)≤1.\hspace{-30.4em}{C}_{03}{e}^{-\sigma t}\left(1+\Vert v\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{2}+\Vert {\nabla }^{m}u\left(\tau -t){\Vert }^{p+1})\le 1.By (3.4), it is easy to get to any t≥0t\ge 0, (4.22)−2∫ΩG(x,u)dx≤−2c3∫Ω∣u∣p+1dx+2∫Ωϕ3dx≤2∫Ωϕ3dx.\hspace{-30.4em}-2\mathop{\int }\limits_{\Omega }G\left(x,u){\rm{d}}x\le -2{c}_{3}\mathop{\int }\limits_{\Omega }| u{| }^{p+1}{\rm{d}}x+2\mathop{\int }\limits_{\Omega }{\phi }_{3}{\rm{d}}x\le 2\mathop{\int }\limits_{\Omega }{\phi }_{3}{\rm{d}}x.When ∣ξ∣→∞| \xi | \to \infty , w(ξ)w\left(\xi )at most polynomial growth, C01e−στ∫−∞τeσξ(1+∣θ−τw(ξ)∣2+∣θ−τw(ξ)∣21−2q+∣θ−τw(ξ)∣p+1)dξ≡r00(τ,w).{C}_{01}{e}^{-\sigma \tau }\underset{-\infty }{\overset{\tau }{\int }}{e}^{\sigma \xi }\left(1+| {\theta }_{-\tau }w\left(\xi ){| }^{2}+| {\theta }_{-\tau }w\left(\xi ){| }^{\tfrac{2}{1-2q}}+| {\theta }_{-\tau }w\left(\xi ){| }^{p+1}){\rm{d}}\xi \equiv {r}_{00}\left(\tau ,w).We get from (4.18) and (4.21) that (4.23)‖v(τ,τ−t,θ−τw,vτ−t)‖Lμ22+‖∇mu(τ,τ−t,θ−τw,vτ−t)‖2≤C041+2α−1λ11−me−στ∫−∞τeσξ‖f(⋅,ξ)‖Lμ22dξ+r00(τ,w)≡r10(τ,w),\begin{array}{l}\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{L}_{\mu }^{2}}^{2}+\Vert {\nabla }^{m}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}\le {C}_{04}\left(1+2{\alpha }^{-1}{\lambda }_{1}^{1-m}{e}^{-\sigma \tau }\underset{-\infty }{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \xi }\Vert f\left(\cdot ,\hspace{0.33em}\xi ){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}\xi \right)+{r}_{00}\left(\tau ,w)\equiv {r}_{10}\left(\tau ,w),\end{array}and r10(τ,w){r}_{10}\left(\tau ,w)is bounded.□Taking the inner product of (3.8) with (−Δ)kv,k=1,2,…,m−1{\left(-\Delta )}^{k}v,k=1,2,\ldots ,m-1in L2(Ω){L}^{2}\left(\Omega ), we find that (4.24)12ddt‖∇kv‖2=ε‖∇kv‖2−(a(x)(−Δ)mv,(−Δ)kv)+ε(a(x)(−Δ)mu,(−Δ)kv)−ε2(u,(−Δ)kv)−(b(x)M(‖∇mu‖2)(−Δ)mu,(−Δ)kv)−(g(x,u),(−Δ)kv)+(f(x,t),(−Δ)kv)+εw(t)(h,(−Δ)kv)−w(t)(a(x)(−Δ)mh,(−Δ)kv).\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}v{\Vert }^{2}& =& \varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}-\left(a\left(x){\left(-\Delta )}^{m}v,{\left(-\Delta )}^{k}v)+\varepsilon \left(a\left(x){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)\\ & & \phantom{\rule[-0.75em]{}{0ex}}-{\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}v)-\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)-\left(g\left(x,u),{\left(-\Delta )}^{k}v)\\ & & +(f\left(x,t),{\left(-\Delta )}^{k}v)+\varepsilon w\left(t)\left(h,{\left(-\Delta )}^{k}v)-w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v).\end{array}For each term on the right-hand side of (4.24): (4.25)(a(x)(−Δ)mv,(−Δ)kv)=(a(x)∇m+kv,∇m+kv)+∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+kv,\left(a\left(x){\left(-\Delta )}^{m}v,{\left(-\Delta )}^{k}v)=\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)+\left(\mathop{\sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v\right),(4.26)ε(a(x)(−Δ)mu,(−Δ)kv)=ε(a(x)∇m+ku,∇m+kv)+ε∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+ku=ε12ddt(a(x)∇m+ku,∇m+ku)+ε2(a(x)∇m+ku,∇m+ku)−εw(t)(a(x)∇m+ku,∇m+kh)+ε∑i=1m−kCm−ki∇m+k−iv∇ia(x),∇m+ku,\hspace{2.35em}\begin{array}{l}\varepsilon \left(a\left(x){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)=\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}v)+\varepsilon \left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u\right)\\ \hspace{9.82em}=\hspace{-0.25em}\varepsilon \frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{10.82em}-\hspace{0.33em}\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+\hspace{-0.25em}\varepsilon \left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u\right),\end{array}(4.27)ε2(u,(−Δ)kv)=ε2(u,(−Δ)k(ut+εu−hw(t)))=ε22ddt‖∇ku‖2+ε3‖∇ku‖2−ε2w(t)(∇ku,∇kh),{\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}v)={\varepsilon }^{2}\left(u,{\left(-\Delta )}^{k}\left({u}_{t}+\varepsilon u-hw\left(t)))=\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}-{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h),(4.28)(b(x)M(‖∇mu‖2)(−Δ)mu,(−Δ)kv)=b0M(‖∇mu‖2)(a(x)∇m+ku,∇m+kv)+M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku=b02M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−b0M(‖∇mu‖2)w(t)(a(x)∇m+ku,∇m+kh)+M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku,\hspace{-39.1em}\begin{array}{l}\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{\left(-\Delta )}^{k}v)\\ \hspace{1.0em}={b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}v)+M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right)\\ \hspace{1.0em}=\frac{{b}_{0}}{2}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{2.0em}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right),\end{array}(4.29)(g(x,u),(−Δ)kv)=(∇xkg(x,u),∇kv)≤∫Ω(c5∣u∣p+ϕ5(x))∇kvdx≤c5∫Ω∣u∣p∇kvdx+∫Ωϕ5(x)∇kvdx≤c5‖u‖L2pp‖∇kv‖+‖ϕ5(x)‖‖∇kv‖≤a008‖∇m+kv‖2+Ck1(r01p(τ,w)+‖ϕ5(x)‖2),\hspace{-39.1em}\begin{array}{rcl}\left(g\left(x,u),{\left(-\Delta )}^{k}v)& =& \left({\nabla }_{x}^{k}g\left(x,u),{\nabla }^{k}v)\le \left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }\left({c}_{5}| u{| }^{p}+{\phi }_{5}\left(x)){\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|\\ & \le & {c}_{5}\left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }| u{| }^{p}{\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|+\left|\hspace{-0.25em}\mathop{\displaystyle \int }\limits_{\Omega }{\phi }_{5}\left(x){\nabla }^{k}v{\rm{d}}x\hspace{-0.25em}\right|\\ & \le & {c}_{5}\Vert u{\Vert }_{{L}^{2p}}^{p}\Vert {\nabla }^{k}v\Vert +\Vert {\phi }_{5}\left(x)\Vert \Vert {\nabla }^{k}v\Vert \\ & \le & \frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}+{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2}),\end{array}(4.30)(f(x,t),(−Δ)kv)=(∇kf(x,t),∇kv)≤‖∇kf(x,t)‖‖∇kv‖≤a008‖∇m+kv‖2+2λ1−ma00‖∇kf(x,t)‖2,\hspace{-38.92em}(f\left(x,t),{\left(-\Delta )}^{k}v)=\left({\nabla }^{k}f\left(x,t),{\nabla }^{k}v)\le \Vert {\nabla }^{k}f\left(x,t)\Vert \Vert {\nabla }^{k}v\Vert \le \frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2},moreover, (4.31)(∇m+k−iv∇ia(x),∇m+kv)≤ai‖∇m+k−iv‖‖∇m+kv‖,i=1,2,…,m−k,ai=‖∇ia(x)‖∞.\left({\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v)\le {a}_{i}\Vert {\nabla }^{m+k-i}v\Vert \Vert {\nabla }^{m+k}v\Vert ,\hspace{1em}i=1,2,\ldots ,m-k,{a}_{i}=\Vert {\nabla }^{i}a\left(x){\Vert }_{\infty }.According to the interpolation inequality, we have ‖∇m+k−iv‖≤Ci‖∇m+kv‖αi‖v‖1−αi,αi=m+k−im+k,\Vert {\nabla }^{m+k-i}v\Vert \le {C}_{i}\Vert {\nabla }^{m+k}v{\Vert }^{{\alpha }_{i}}\Vert v{\Vert }^{1-{\alpha }_{i}},\hspace{1em}{\alpha }_{i}=\frac{m+k-i}{m+k},then (4.32)(Cm−ki∇m+k−iv∇ia(x),∇m+kv)≤Cm−kiCiai‖v‖1−αi‖∇m+kv‖1+αi≤a008(m−k)‖∇m+kv‖2+1−αi2a004(1−αi)(m−k)1+αi1−αi(Cm−kiCiai)21−αi‖v‖2,\begin{array}{lcl}\left({C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}v)& \le & {C}_{m-k}^{i}{C}_{i}{a}_{i}\Vert v{\Vert }^{1-{\alpha }_{i}}\Vert {\nabla }^{m+k}v{\Vert }^{1+{\alpha }_{i}}\\ & \le & \frac{{a}_{00}}{8\left(m-k)}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\frac{1-{\alpha }_{i}}{2}{\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{C}_{i}{a}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2},\end{array}(4.33)(Cm−ki∇m+k−iv∇ia(x),∇m+ku)≤a00b0M08(m−k)‖∇m+ku‖2+2(m−k)a00b0M0(Cm−kiai)2‖∇m+k−iv‖2≤a00b0M08(m−k)‖∇m+ku‖2+a008(m−k)‖∇m+kv‖2+(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2,\hspace{-41.2em}\begin{array}{lcl}\left({C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}a\left(x),{\nabla }^{m+k}u)& \le & \frac{{a}_{00}{b}_{0}{M}_{0}}{8\left(m-k)}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i})}^{2}\Vert {\nabla }^{m+k-i}v{\Vert }^{2}\\ & \hspace{0.425em}\le \hspace{-0.25em}& \frac{{a}_{00}{b}_{0}{M}_{0}}{8\left(m-k)}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{{a}_{00}}{8\left(m-k)}\Vert {\nabla }^{m+k}v{\Vert }^{2}+\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2},\end{array}(4.34)b0M(‖∇mu‖2)w(t)(a(x)∇m+ku,∇m+kh)≤b0a0M0(1+‖∇mu‖2q)∣w(t)∣‖∇m+ku‖‖∇m+kh‖≤εb0a00M08‖∇m+ku‖2+2ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2,\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)\\ \hspace{1.0em}\le {b}_{0}{a}_{0}{M}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})| w\left(t)| \Vert {\nabla }^{m+k}u\Vert \Vert {\nabla }^{m+k}h\Vert \\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+2{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2},\end{array}(4.35)M(‖∇mu‖2)∑i=1m−kCm−ki∇m+k−iv∇ib(x),∇m+ku≤M0b0(1+‖∇mu‖2q)∑i=1m−kCm−kiai‖∇m+k−iv‖‖∇m+ku‖≤εb0a00M08‖∇m+ku‖2+ε−1a00−1b0M0∑i=1m−k(Cm−ki(1+‖∇mu‖2q)ai)2‖∇m+k−iv‖2≤εb0a00M08‖∇m+ku‖2+a008‖∇m+kv‖2+(ε−1a00−1b0M0)11−αi∑i=1m−k(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2.\begin{array}{l}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}v{\nabla }^{i}b\left(x),{\nabla }^{m+k}u\right)\\ \hspace{1.0em}\le {M}_{0}{b}_{0}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{a}_{i}\Vert {\nabla }^{m+k-i}v\Vert \Vert {\nabla }^{m+k}u\Vert \\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}){a}_{i})}^{2}\Vert {\nabla }^{m+k-i}v{\Vert }^{2}\\ \hspace{1.0em}\le \frac{\varepsilon {b}_{0}{a}_{00}{M}_{0}}{8}\Vert {\nabla }^{m+k}u{\Vert }^{2}+\frac{{a}_{00}}{8}\Vert {\nabla }^{m+k}v{\Vert }^{2}\\ \hspace{2.0em}+\hspace{0.25em}{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}.\end{array}By (4.25)–(4.30) and (4.32)–(4.35), we get (4.36)ddt[‖∇kv‖2−ε(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2(a(x)∇m+kv,∇m+kv)−4+ε4a00‖∇m+kv‖2−2ε‖∇kv‖2+2εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)+2ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}-\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]+{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{4+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-2\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\end{array}+2∑i=1m−k(ε−1a00−1b0M0)11−αi(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)−2εw(t)(a(x)∇m+ku,∇m+kh)+2ε2w(t)(∇ku,∇kh)+2εw(t)((−Δ)ku,h)−2w(t)(a(x)(−Δ)mh,(−Δ)kv).\begin{array}{l}\hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})-2\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)+2{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h)\\ \hspace{2.0em}+2\varepsilon w\left(t)\left({\left(-\Delta )}^{k}u,h)-2w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v).\end{array}Using the Cauchy-Schwarz inequality,Young’s inequality and Holder’s inequality, etc. we have (4.37)2εw(t)(a(x)∇m+ku,∇m+kh)≤ε2a00‖∇m+ku‖2+a00−1a02∣w(t)∣2‖∇m+kh‖2,2\varepsilon w\left(t)\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}h)\le {\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2},(4.38)2ε2w(t)(∇ku,∇kh)≤ε3‖∇ku‖2+ε∣w(t)∣2‖∇kh‖2,\hspace{-21.6em}2{\varepsilon }^{2}w\left(t)\left({\nabla }^{k}u,{\nabla }^{k}h)\le {\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2},(4.39)2εw(t)((−Δ)ku,h)≤ε‖∇kv‖2+ε∣w(t)∣2‖∇kh‖2,\hspace{-21.6em}2\varepsilon w\left(t)\left({\left(-\Delta )}^{k}u,h)\le \varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2},(4.40)2w(t)(a(x)(−Δ)mh,(−Δ)kv)=2w(t)(a(x)∇m+kv,∇m+kh)+2w(t)∑i=1m−kCm−ki∇m+k−i∇ia(x),∇m+kh≤a004‖∇m+kv‖2+4a00−1a02∣w(t)∣2‖∇m+k‖2+a004‖∇m+kv‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2≤a002‖∇m+kv‖2+4a00−1a02∣w(t)∣2‖∇m+k‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}2w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}v)\\ \hspace{1.0em}=2w\left(t)\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}h)+2w\left(t)\left(\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{C}_{m-k}^{i}{\nabla }^{m+k-i}{\nabla }^{i}a\left(x),{\nabla }^{m+k}h\right)\\ \hspace{1.0em}\le \frac{{a}_{00}}{4}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+\frac{{a}_{00}}{4}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}\\ \hspace{1.0em}\le \frac{{a}_{00}}{2}\Vert {\nabla }^{m+k}v{\Vert }^{2}+4{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}Substitute (4.37)–(4.40) into (4.36) to obtain (4.41)ddt[‖∇kv‖2−ε(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2(a(x)∇m+kv,∇m+kv)−6+ε4a00‖∇m+kv‖2−3ε‖∇kv‖2+2εb0M(‖∇mu‖2)(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)−ε2a00‖∇m+ku‖2+ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2+2∑i=1m−k(1−αi)(ε−1a00−1b0M0)11−αia008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+k‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}-\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]+{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}+2\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\\ \hspace{1.0em}-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}{\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}When ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge 0, b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥ddt(b0M0(a(x)∇m+ku,∇m+ku))+2εb0M0(a(x)∇m+ku,∇m+ku);\hspace{-34em}\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\right)\\ \hspace{1.0em}\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}{M}_{0}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u))+2\varepsilon {b}_{0}{M}_{0}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u);\end{array}else b0M(‖∇mu‖2)ddt(a(x)∇m+ku,∇m+ku)+2ε(a(x)∇m+ku,∇m+ku)≥ddt(b0C(‖∇mu‖2)(a(x)∇m+ku,∇m+ku))+2εb0C(‖∇mu‖2)(a(x)∇m+ku,∇m+ku),\begin{array}{l}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+2\varepsilon \left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\right)\\ \hspace{1.0em}\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u))+2\varepsilon {b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u),\end{array}then (4.41) is transformed into (4.42)ddt[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+2(a(x)∇m+kv,∇m+kv)−6+ε4a00‖∇m+kv‖2−3ε‖∇kv‖2+2εb0M0(C(‖∇mu‖2))(a(x)∇m+ku,∇m+ku)−3εa00b0M04‖∇m+ku‖2−2ε2(a(x)∇m+ku,∇m+ku)−ε2a00‖∇m+ku‖2+ε3‖∇ku‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖v‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖v‖2+2∑i=1m−k(1−αi)(ε−1a00−1b0M0)11−αia008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖v‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λ1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+k‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}v{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}u{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+2\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}{\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert v{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}By 2(a(x)∇m+kv,∇m+kv)≥2a00‖∇m+kv‖2,(a(x)∇m+ku,∇m+ku)≥a00‖∇m+ku‖22\left(a\left(x){\nabla }^{m+k}v,{\nabla }^{m+k}v)\ge 2{a}_{00}\Vert {\nabla }^{m+k}v{\Vert }^{2},\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge {a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}, (4.1), we have (4.43)ddt[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]+σ1[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]≤Ck2(1+∣w(t)∣2)+2λ1−ma00‖∇kf(x,t)‖2.\hspace{-34.3em}\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1em}+{\sigma }_{1}\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1em}\le {C}_{k2}\left(1+| w\left(t){| }^{2})+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}.\end{array}When k=mk=m, (4.43) is also true, which will not be detailed here.Using the Gronwall inequality to integrate (4.43) over [τ−t,τ]\left[\tau -t,\tau ]and replacing wwby θ−τw{\theta }_{-\tau }wwe obtain (4.44)eσ1τ[‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2]≤eσ1(τ−t)[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]+∫τ−tτeσ1ξCk2(1+∣w(ξ)∣2)dξ+2λ1−ma00∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ,\begin{array}{l}{e}^{{\sigma }_{1}\tau }\left[\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}]\\ \hspace{1.0em}\le {e}^{{\sigma }_{1}\left(\tau -t)}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]+\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi ,\end{array}moreover, (4.45)‖∇kv‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku,∇m+ku)+ε2‖∇ku‖2≤e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk2(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}v{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)+{\varepsilon }^{2}\Vert {\nabla }^{k}u{\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi .\end{array}Since (u(τ−t),v(τ−t))∈Bk(τ−t,θ−τw)\left(u\left(\tau -t),v\left(\tau -t))\in {B}_{k}\left(\tau -t,{\theta }_{-\tau }w), when t→+∞t\to +\infty , (4.46)e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+ku(τ−t),∇m+ku(τ−t))+ε2‖∇ku(τ−t)‖2]≤e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)a0‖∇m+ku(τ−t)‖2+ε2‖∇ku(τ−t)‖2]→0,\begin{array}{l}{e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}u\left(\tau -t),{\nabla }^{m+k}u\left(\tau -t))+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon ){a}_{0}\Vert {\nabla }^{m+k}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\to 0,\end{array}then there exists Tk=Tk(τ,w,Bk){T}_{k}={T}_{k}\left(\tau ,w,{B}_{k})such that for all t≥Tkt\ge {T}_{k}(4.47)e−σ1t[‖∇kv(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)a0‖∇m+ku(τ−t)‖2+ε2‖∇ku(τ−t)‖2]≤1.{e}^{-{\sigma }_{1}t}\left[\Vert {\nabla }^{k}v\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon ){a}_{0}\Vert {\nabla }^{m+k}u\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{k}u\left(\tau -t){\Vert }^{2}]\le 1.When ∣ξ∣→∞w(ξ)| \xi | \to \infty \hspace{0.33em}w\left(\xi )at most polynomial growth, e−σ1τ∫−∞τeσ1ξCk2(1+∣w(ξ)∣2)dξ≡r0k(τ,w).{e}^{-{\sigma }_{1}\tau }\underset{-\infty }{\overset{\tau }{\int }}{e}^{{\sigma }_{1}\xi }{C}_{k2}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi \equiv {r}_{0k}\left(\tau ,w).We get from (4.45)–(4.47), (a(x)∇m+ku,∇m+ku)≥a00‖∇m+ku‖2\left(a\left(x){\nabla }^{m+k}u,{\nabla }^{m+k}u)\ge {a}_{00}\Vert {\nabla }^{m+k}u{\Vert }^{2}that (4.48)‖∇kv(τ,τ−t,θ−τw,vτ−t)‖2+‖∇m+ku(τ,τ−t,θ−τw,vτ−t)‖2≤Ck31+2λ1−ma00e−σ1τ∫−∞τeσ1ξ‖∇kf(x,ξ)‖2dξ+r0k(τ,w)≡r1k(τ,w),\begin{array}{l}\Vert {\nabla }^{k}v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}+\Vert {\nabla }^{m+k}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }^{2}\\ \hspace{1.0em}\le {C}_{k3}\left(1+\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{-\infty }{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi \right)+{r}_{0k}\left(\tau ,w)\equiv {r}_{1k}\left(\tau ,w),\end{array}and r1k(τ,w){r}_{1k}\left(\tau ,w)are bounded.Lemma 4.1 is derived from (4.23) and (4.48).Lemma 4.1 is proved.Considering the eigenvalue problem (−Δ)m+ku=λm+ku,u∣Γ=0,{\left(-\Delta )}^{m+k}u={\lambda }^{m+k}u,u\hspace{-0.25em}{| }_{\Gamma }=0,the problem (3.8) has a family of eigenfunctions {ej}j=1∞{\left\{{e}_{j}\right\}}_{j=1}^{\infty }with the eigenvalues {λj}j=1∞:λ1≤λ2≤⋯≤λj→∞(j→∞){\left\{{\lambda }_{j}\right\}}_{j=1}^{\infty }:{\lambda }_{1}\le {\lambda }_{2}\hspace{0.33em}\le \cdots \le {\lambda }_{j}\to \infty \left(j\to \infty ), such that {ej}j=1∞{\left\{{e}_{j}\right\}}_{j=1}^{\infty }is an orthonormal basis of Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ). Given nnlet Qn=span{e1,⋅,en}{Q}_{n}={\rm{span}}\left\{{e}_{1},\cdot ,{e}_{n}\right\}and Pn:Vk(Ω)→Qn{P}_{n}\hspace{0.25em}:{V}_{k}\left(\Omega )\to {Q}_{n}be the projection operator.Lemma 4.2Suppose MMsatisfies (M),h(x)∈Vm+k(Ω),k=0,1,…,m\left(M),h\left(x)\in {V}_{m+k}\left(\Omega ),k=0,1,\ldots ,m(3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1})and for ∀η>0,τ∈R,w∈Ω1,Bk={Bk(τ,w):τ∈R,w∈Ω1}∈Dk\forall \hspace{-0.25em}\eta \gt 0,\tau \in R,w\in {\Omega }_{1},{B}_{k}=\left\{{B}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}, there exists Tk=Tk(τ,w,Bk,ηk)>0,Nk=Nk(τ,w,ηk)≥0{T}_{k}={T}_{k}\left(\tau ,w,{B}_{k},{\eta }_{k})\gt 0,{N}_{k}={N}_{k}\left(\tau ,w,{\eta }_{k})\ge 0such that the solution of (3.8) satisfies for t≥Tk,n≥Nkt\ge {T}_{k},n\ge {N}_{k}‖(I−Pn)v(τ,τ−t,θ−τw)‖Vk2+‖(I−Pn)u(τ,τ−t,θ−τw)‖Vm+k2≤ηk.\Vert \left(I-{P}_{n})v\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{V}_{k}}^{2}+\Vert \left(I-{P}_{n})u\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{V}_{m+k}}^{2}\le {\eta }_{k}.ProofLet un,1=Pnu,un,2=u−un,1,vn,1=Pnv,vn,2=v−vn,1{u}_{n,1}={P}_{n}u,{u}_{n,2}=u-{u}_{n,1},{v}_{n,1}={P}_{n}v,{v}_{n,2}=v-{v}_{n,1}. Applying (I−Pn)\left(I-{P}_{n})to the second equation of (3.8), we obtain (4.49)dvn,2dt=εvn,2−a(x)(−Δ)mvn,2+εa(x)(−Δ)mun,2−ε2un,2−b(x)(I−Pn)M(‖∇m‖2)(−Δ)mu−(I−Pn)(g(x,u)+f(x,t))+εh(x)w(t)−a(x)(−Δ)mh(x)w(t).\begin{array}{l}\frac{{\rm{d}}{v}_{n,2}}{{\rm{d}}t}=\varepsilon {v}_{n,2}-a\left(x){\left(-\Delta )}^{m}{v}_{n,2}+\varepsilon a\left(x){\left(-\Delta )}^{m}{u}_{n,2}-{\varepsilon }^{2}{u}_{n,2}-b\left(x)\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}{\Vert }^{2}){\left(-\Delta )}^{m}u\\ \hspace{3.25em}-\left(I-{P}_{n})\left(g\left(x,u)+f\left(x,t))+\varepsilon h\left(x)w\left(t)-a\left(x){\left(-\Delta )}^{m}h\left(x)w\left(t).\end{array}Taking the inner product of the resulting equation (4.49) with vn,2{v}_{n,2}in Lμ2(Ω){L}_{\mu }^{2}\left(\Omega ), we have (4.50)12ddt‖vn,2‖Lμ22=ε‖vn,2‖Lμ22−‖∇mvn,2‖2+ε((−Δ)mun,2,vn,2)−ε2(un,2,vn,2)Lμ2−b0((I−Pn)M(‖∇mu‖2)(−Δ)mu,vn,2)−((I−Pn)g(x,u),vn,2)Lμ2+((I−Pn)f(x,t),vn,2)Lμ2+εw(t)(h(x),vn,2)Lμ2−w(t)((−Δ)mh,vn,2).\begin{array}{rcl}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}& =& \varepsilon \Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}-\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},{v}_{n,2})-{\varepsilon }^{2}{\left({u}_{n,2},{v}_{n,2})}_{{L}_{\mu }^{2}}\\ & & -{b}_{0}\left(\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{v}_{n,2})-{\left(\left(I-{P}_{n})g\left(x,u),{v}_{n,2})}_{{L}_{\mu }^{2}}\\ & & +{\left(\left(I-{P}_{n})f\left(x,t),{v}_{n,2})}_{{L}_{\mu }^{2}}+\varepsilon w\left(t){\left(h\left(x),{v}_{n,2})}_{{L}_{\mu }^{2}}-w\left(t)\left({\left(-\Delta )}^{m}h,{v}_{n,2}).\end{array}Applying I−PnI-{P}_{n}to the first equation of (3.8), we get (4.51)vn,2=dun,2dt+εun,2−hw(t).{v}_{n,2}=\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t).For the third and fourth terms on the right-hand side of (4.49), we obtain (4.52)ε((−Δ)mun,2,vn,2)−ε2(un,2,vn,2)Lμ2=ε((−Δ)mun,2,dun,2dt+εun,2−hw(t))−ε2(un,2,dun,2dt+εun,2−hw(t))Lμ2=ε2ddt‖∇mun,2‖2+ε2‖∇mun,2‖2−εw(t)((−Δ)mun,2,h)−ε22ddt‖un,2‖Lμ22−ε3‖un,2‖Lμ22+ε2w(t)(un,2,h)Lμ2,\begin{array}{l}\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},{v}_{n,2})-{\varepsilon }^{2}{\left({u}_{n,2},{v}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\varepsilon \left({\left(-\Delta )}^{m}{u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))-\hspace{-0.25em}{\varepsilon }^{2}{\left({u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\hspace{0.25em}\frac{\varepsilon }{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}-\varepsilon w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h)\\ \hspace{2.0em}-\frac{{\varepsilon }^{2}}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}-{\varepsilon }^{3}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{2}w\left(t){\left({u}_{n,2},h)}_{{L}_{\mu }^{2}},\end{array}for the fifth term on the right-hand side of (4.49), we have (4.53)b0((I−Pn)M(‖∇mu‖2)(−Δ)mu,vn,2)=b0(M(‖∇mu‖2)(−Δ)mun,2,dun,2dt+εun,2−hw(t))=12b0M(‖∇mu‖2)ddt‖∇mun,2‖2+εb0M(‖∇mu‖2)∣∇mun,2‖2−b0M(‖∇mu‖2)w(t)((−Δ)mun,2,h),\begin{array}{l}{b}_{0}\left(\left(I-{P}_{n})M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}u,{v}_{n,2})\\ \hspace{1.0em}={b}_{0}\left(M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}{u}_{n,2},\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))\\ \hspace{1.0em}=\frac{1}{2}{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\varepsilon {b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})| {\nabla }^{m}{u}_{n,2}{\Vert }^{2}-{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h),\end{array}for the sixth term on the right-hand side of (4.49), we have (4.54)((I−Pn)g(x,u),vn,2)Lμ2=((I−Pn)g(x,u),dun,2dt+εun,2−hw(t))Lμ2=ddt((I−Pn)g(x,u),un,2)Lμ2−((I−Pn)gu(x,u)ut,un,2)Lμ2+ε((I−Pn)g(x,u),un,2)Lμ2−((I−Pn)g(x,u),hw(t))Lμ2,\begin{array}{l}{\left(\left(I-{P}_{n})g\left(x,u),{v}_{n,2})}_{{L}_{\mu }^{2}}=\hspace{-0.25em}{\left(\left(I-{P}_{n})g\left(x,u),\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}=\frac{{\rm{d}}}{{\rm{d}}t}{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}-{\left(\left(I-{P}_{n}){g}_{u}\left(x,u){u}_{t},{u}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{2.0em}+\hspace{0.25em}\varepsilon {\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}-{\left(\left(I-{P}_{n})g\left(x,u),hw\left(t))}_{{L}_{\mu }^{2}},\end{array}for the seventh, eighth, and ninth terms on the right-hand side of (4.49), we have (4.55)((I−Pn)f(x,t),vn,2)Lμ2≤14‖∇mvn,2‖2+1αn+1λn+1m−1‖(I−Pn)f(x,t)‖Lμ22,{\left(\left(I-{P}_{n})f\left(x,t),{v}_{n,2})}_{{L}_{\mu }^{2}}\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\frac{1}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{P}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2},(4.56)εw(t)(h(x),vn,2)Lμ2≤14‖∇mvn,2‖2+ε∣w(t)∣2αn+1λn+1m−1‖h(x)‖Lμ22,\hspace{-24.5em}\varepsilon w\left(t){\left(h\left(x),{v}_{n,2})}_{{L}_{\mu }^{2}}\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\frac{\varepsilon | w\left(t){| }^{2}}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2},(4.57)w(t)((−Δ)mh,vn,2)≤14‖∇mvn,2‖2+ε∣w(t)∣2‖∇mh(x)‖2.\hspace{-24.5em}w\left(t)\left({\left(-\Delta )}^{m}h,{v}_{n,2})\le \frac{1}{4}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{m}h\left(x){\Vert }^{2}.When ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥0\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\ge 0(4.58)b0M(‖∇mu‖2)ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥ddt(b0M0‖∇mun,2‖2)+2εb0M0‖∇mvn,2‖2,{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\right)\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}{M}_{0}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2})+2\varepsilon {b}_{0}{M}_{0}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2},else (4.59)b0M(‖∇mu‖2)ddt‖∇mun,2‖2+2ε‖∇mvn,2‖2≥ddt(b0C(‖∇mu‖2)‖∇mun,2‖2)+2εb0C(‖∇mu‖2)‖∇mvn,2‖2.{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})\left(\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\varepsilon \Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}\right)\ge \frac{{\rm{d}}}{{\rm{d}}t}\left({b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2})+2\varepsilon {b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}.By using Young’s inequality and Holder’s inequality, we can get (4.60)2b0M(‖∇mu‖2)w(t)((−Δ)mun,2,h)≤2b0M(‖∇mu‖2)w(t)‖∇mun,2‖∣w(t)∣‖∇mh‖≤εb0M02‖∇mun,2‖2+2b0C(‖∇mu‖2)εM0∣w(t)∣2‖∇mh‖2,\begin{array}{lcl}2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h)& \le \hspace{-0.25em}& 2{b}_{0}M\left(\Vert {\nabla }^{m}u{\Vert }^{2})w\left(t)\Vert {\nabla }^{m}{u}_{n,2}\Vert | w\left(t)| \Vert {\nabla }^{m}h\Vert \\ & \le & \frac{\varepsilon {b}_{0}{M}_{0}}{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2},\end{array}(4.61)2ε2w(t)(h(x),un,2)Lμ2−2εw(t)((−Δ)mun,2,h(x))≤ε3‖un,2‖Lμ22+ε2‖∇mun,2‖2+ε∣w(t)∣2‖h‖Lμ22+∣w(t)∣2‖∇mh‖2.\hspace{-35.35em}\begin{array}{l}2{\varepsilon }^{2}w\left(t){\left(h\left(x),{u}_{n,2})}_{{L}_{\mu }^{2}}-2\varepsilon w\left(t)\left({\left(-\Delta )}^{m}{u}_{n,2},h\left(x))\\ \hspace{1.0em}\le {\varepsilon }^{3}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+{\varepsilon }^{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\varepsilon | w\left(t){| }^{2}\Vert h{\Vert }_{{L}_{\mu }^{2}}^{2}+| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}.\end{array}By (3.5), we have (4.62)2((I−Pn)gu(x,u)ut,un,2)Lμ2≤2‖ϕ4‖Lμ4‖ut‖Lμ2‖un,2‖Lμ4+2c4‖ut‖Lμ2‖u‖Lμ2pp−1‖un,2‖Lμ2p≤εb0M02‖∇mun,2‖2+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2,\begin{array}{l}2{\left(\left(I-{P}_{n}){g}_{u}\left(x,u){u}_{t},{u}_{n,2})}_{{L}_{\mu }^{2}}\le \hspace{-0.25em}2\Vert {\phi }_{4}{\Vert }_{{L}_{\mu }^{4}}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{4}}+2{c}_{4}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}\Vert u{\Vert }_{{L}_{\mu }^{2p}}^{p-1}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2p}}\le \hspace{-0.25em}\frac{\varepsilon {b}_{0}{M}_{0}}{2}\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2},\end{array}(4.63)2w(t)((I−Pn)g(x,u),h)Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2c1∣w(t)∣‖u‖Lμ2pp‖h‖Lμ2≤2∣w(t)∣‖ϕ1‖Lμ2‖h‖Lμ2+2C07∣w(t)∣‖∇mu‖p‖h‖Lμ2.\hspace{-38.78em}\begin{array}{l}2w\left(t){\left(\left(I-{P}_{n})g\left(x,u),h)}_{{L}_{\mu }^{2}}\le \hspace{-0.25em}2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{c}_{1}| w\left(t)| \Vert u{\Vert }_{{L}_{\mu }^{2p}}^{p}\Vert h{\Vert }_{{L}_{\mu }^{2}}\le 2| w\left(t)| \Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{07}| w\left(t)| \Vert {\nabla }^{m}u{\Vert }^{p}\Vert h{\Vert }_{{L}_{\mu }^{2}}.\end{array}By substituting (4.52)–(4.63) into (4.50), we have (4.64)ddt[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]+12‖∇mvn,2‖2−2ε‖vn,2‖Lμ22+(2εM(‖∇mu‖2)−εM0)b0‖∇mu‖2−3ε2‖∇mu‖2+ε3‖u‖Lμ22+2ε((I−Pn)g(x,u),un,2)Lμ2≤1+2b0C(‖∇mu‖2)εM0+2ε∣w(t)∣2‖∇mh‖2+ε+2εαn+1λn+1m−1∣w(t)∣2‖h(x)‖Lμ22+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2+(2‖ϕ1‖Lμ2‖h‖Lμ2+2C08‖∇mu‖p)∣w(t)∣‖h‖Lμ2+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}+\frac{1}{2}\Vert {\nabla }^{m}{v}_{n,2}{\Vert }^{2}-2\varepsilon \Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left(2\varepsilon M\left(\Vert {\nabla }^{m}u{\Vert }^{2})-\varepsilon {M}_{0}){b}_{0}\Vert {\nabla }^{m}u{\Vert }^{2}\\ \hspace{1.0em}-\hspace{0.25em}3{\varepsilon }^{2}\Vert {\nabla }^{m}u{\Vert }^{2}+{\varepsilon }^{3}\Vert u{\Vert }_{{L}_{\mu }^{2}}^{2}+2\varepsilon {\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le \hspace{-0.25em}\left(1+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}+2\varepsilon \right)| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\left(\varepsilon +\frac{2\varepsilon }{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\right)| w\left(t){| }^{2}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{2.0em}+\hspace{0.25em}\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+\hspace{0.25em}{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2}+\left(2\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{08}\Vert {\nabla }^{m}u{\Vert }^{p})| w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}\hspace{2em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}.\end{array}Because, when N=1,2,then1≤p<+∞N=1,2,\text{then}\hspace{0.25em}1\le p\lt +\infty ; when N=3,4,then1≤p<NN−2N=3,4,\text{then}\hspace{0.25em}1\le p\lt \frac{N}{N-2}. Moreover, when n→∞,λn→∞n\to \infty ,{\lambda }_{n}\to \infty so given η0>0{\eta }_{0}\gt 0, there exists N01=N01(η0)≥1{N}_{01}={N}_{01}\left({\eta }_{0})\ge 1for n≥N01n\ge {N}_{01}(4.65)1+2b0C(‖∇mu‖2)εM0+2ε∣w(t)∣2‖∇mh‖2+ε+2εαn+1λn+1m−1∣w(t)∣2‖h(x)‖Lμ22+C05εb0M0λn+11−m‖ut‖Lμ22+C06λn+11−m‖ut‖Lμ22‖∇mu‖2p−2+(2‖ϕ1‖Lμ2‖h‖Lμ2+2C08‖∇mu‖p)∣w(t)∣‖h‖Lμ2+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22≤C09η0(1+∣w(t)∣2+‖ut‖Lμ26+‖∇mu‖6)+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22.\begin{array}{l}\left(1+2\frac{{b}_{0}C\left(\Vert {\nabla }^{m}u{\Vert }^{2})}{\varepsilon {M}_{0}}+2\varepsilon \right)| w\left(t){| }^{2}\Vert {\nabla }^{m}h{\Vert }^{2}+\left(\varepsilon +\frac{2\varepsilon }{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\right)| w\left(t){| }^{2}\Vert h\left(x){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+\frac{{C}_{05}}{\varepsilon {b}_{0}{M}_{0}}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}+{C}_{06}{\lambda }_{n+1}^{1-m}\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{2}\Vert {\nabla }^{m}u{\Vert }^{2p-2}\\ \hspace{1.0em}+\left(2\Vert {\phi }_{1}{\Vert }_{{L}_{\mu }^{2}}\Vert h{\Vert }_{{L}_{\mu }^{2}}+2{C}_{08}\Vert {\nabla }^{m}u{\Vert }^{p})| w\left(t)| \Vert h{\Vert }_{{L}_{\mu }^{2}}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}\le {C}_{09}{\eta }_{0}\left(1+| w\left(t){| }^{2}+\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u{\Vert }^{6})+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2}.\end{array}Then, there is an appropriate positive constant σ2{\sigma }_{2}so that (4.64) can be reduced to (4.66)ddt[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]+σ2[‖vn,2‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2‖2+ε2‖un,2‖Lμ22+2((I−Pn)g(x,u),un,2)Lμ2]≤C09η0(1+∣w(t)∣2+‖ut‖Lμ26+‖∇mu‖6)+2αn+1λn+1m−1‖(I−pn)f(x,t)‖Lμ22,\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}+{\sigma }_{2}\left[\Vert {v}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}{\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2})}_{{L}_{\mu }^{2}}]\\ \hspace{1.0em}\le {C}_{09}{\eta }_{0}\left(1+| w\left(t){| }^{2}+\Vert {u}_{t}{\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u{\Vert }^{6})+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\Vert \left(I-{p}_{n})f\left(x,t){\Vert }_{{L}_{\mu }^{2}}^{2},\end{array}integrating (4.66) over (τ−t,τ)\left(\tau -t,\tau )with t≥0t\ge 0we get for all n≥N01n\ge {N}_{01}(4.67)‖vn,2(τ,τ−t,w)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,w)‖2+ε2‖un,2(τ,τ−t,w)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,w))Lμ2\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,w){\Vert }^{2}\\ \hspace{1.0em}+\hspace{0.25em}{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,w))}_{{L}_{\mu }^{2}}\end{array}≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0∫τ−tτeσ(s−t)(1+∣w(s)∣2+‖ut(s,τ−t,w,u1τ)‖Lμ26+‖∇mu(s,τ−t,w,u1τ)‖6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds.\begin{array}{l}\hspace{1.0em}\le \hspace{-0.25em}{e}^{-{\sigma }_{2}t}\left[\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}\hspace{2em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\hspace{2em}+{C}_{09}{\eta }_{0}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\left(1+| w\left(s){| }^{2}+\Vert {u}_{t}\left(s,\tau -t,w,{u}_{1\tau }){\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u\left(s,\tau -t,w,{u}_{1\tau }){\Vert }^{6}){\rm{d}}s+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}s.\end{array}Replacing wwby θ−τw{\theta }_{-\tau }win (4.67) for every t∈R+,τ∈R,w∈Ω1,n≥N01t\in {R}^{+},\tau \in R,w\in {\Omega }_{1},n\ge {N}_{01}, we obtain (4.68)‖vn,2(τ,τ−t,θ−τw)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,θ−τw)‖2+ε2‖un,2(τ,τ−t,θ−τw)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,θ−τw))Lμ2≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0∫τ−tτeσ(s−t)(1+∣w(s)∣2+‖ut(s,τ−t,θ−τw,u1τ)‖Lμ26+‖∇mu(s,τ−t,θ−τw,u1τ)‖6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds.\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}\\ \hspace{1.0em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{2}t}{[}\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}\hspace{2em}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\hspace{2em}+{C}_{09}{\eta }_{0}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}(1+| w\left(s){| }^{2}+\Vert {u}_{t}\left(s,\tau -t,{\theta }_{-\tau }w,{u}_{1\tau }){\Vert }_{{L}_{\mu }^{2}}^{6}+\Vert {\nabla }^{m}u\left(s,\tau -t,{\theta }_{-\tau }w,{u}_{1\tau }){\Vert }^{6}){\rm{d}}s\\ \hspace{2.0em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}{\rm{d}}s.\end{array}From (3.8), f(x,t)f\left(x,t)satisfies (F1),h∈Vm\left({F}_{1}),h\in {V}_{m}and Lemma 4.1, we have (4.69)‖vn,2(τ,τ−t,θ−τw)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ,τ−t,θ−τw)‖2+ε2‖un,2(τ,τ−t,θ−τw)‖Lμ22+2((I−Pn)g(x,u),un,2(τ,τ−t,θ−τw))Lμ2≤e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]+C09η0r003∫τ−tτeσ(s−t)(1+∣w(s)∣2+∣w(s)∣6)ds+2αn+1λn+1m−1∫τ−tτeσ(s−t)‖(I−pn)f(x,s)‖Lμ22ds,\begin{array}{l}\Vert {v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u),{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))}_{{L}_{\mu }^{2}}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{2}t}\left[\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]+{C}_{09}{\eta }_{0}{r}_{00}^{3}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\left(1+| w\left(s){| }^{2}+| w\left(s){| }^{6}){\rm{d}}s\\ \hspace{2.0em}+\frac{2}{{\alpha }_{n+1}{\lambda }_{n+1}^{m-1}}\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{\sigma \left(s-t)}\Vert \left(I-{p}_{n})f\left(x,s){\Vert }_{{L}_{\mu }^{2}}^{2}\hspace{0.33em}{\rm{d}}s,\end{array}by (uτ−t,vτ−t)∈B0(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{0}\left(\tau -t,{\theta }_{-\tau }w), then (4.70)e−σ2t[‖vn,2(τ−t)‖Lμ22+(b0M0(C(‖∇mu‖2))−ε)‖∇mun,2(τ−t)‖2+ε2‖un,2(τ−t)‖Lμ22+2((I−Pn)g(x,u(τ−t)),un,2(τ−t))Lμ2]→0,t→∞.\begin{array}{l}{e}^{-{\sigma }_{2}t}{[}\Vert {v}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\Vert {\nabla }^{m}{u}_{n,2}\left(\tau -t){\Vert }^{2}+{\varepsilon }^{2}\Vert {u}_{n,2}\left(\tau -t){\Vert }_{{L}_{\mu }^{2}}^{2}\\ \hspace{1.0em}+2{\left(\left(I-{P}_{n})g\left(x,u\left(\tau -t)),{u}_{n,2}\left(\tau -t))}_{{L}_{\mu }^{2}}]\to 0,\hspace{1em}t\to \infty .\end{array}Taking the inner product of (4.49) with (−Δ)kvn,2,k=1,2,…,m−1{\left(-\Delta )}^{k}{v}_{n,2},k=1,2,\ldots ,m-1in L2(Ω){L}^{2}\left(\Omega ), we have (4.71)12ddt‖∇kvn,2‖2=ε‖∇kvn,2‖2−(a(x)(−Δ)mvn,2,(−Δ)kvn,2)+ε(a(x)(−Δ)mun,2,(−Δ)kvn,2)−ε2(un,2,(−Δ)kvn,2)−(b(x)M(‖∇mu‖2)(−Δ)mun,2,(−Δ)kvn,2)−((I−Pn)g(x,u),(−Δ)kvn,2)+(f(x,t),(−Δ)kvn,2)+εw(t)(h,(−Δ)kvn,2)−w(t)(a(x)(−Δ)mh,(−Δ)kvn,2).\begin{array}{l}\frac{1}{2}\frac{{\rm{d}}}{{\rm{d}}t}\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}=\hspace{-0.25em}\varepsilon \Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}-\left(a\left(x){\left(-\Delta )}^{m}{v}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})+\varepsilon \left(a\left(x){\left(-\Delta )}^{m}{u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})\\ \hspace{6.67em}-{\varepsilon }^{2}\left({u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})-\left(b\left(x)M\left(\Vert {\nabla }^{m}u{\Vert }^{2}){\left(-\Delta )}^{m}{u}_{n,2},{\left(-\Delta )}^{k}{v}_{n,2})\hspace{6.67em}-\left(\left(I-{P}_{n})g\left(x,u),{\left(-\Delta )}^{k}{v}_{n,2})+(f\left(x,t),{\left(-\Delta )}^{k}{v}_{n,2})+\varepsilon w\left(t)\left(h,{\left(-\Delta )}^{k}{v}_{n,2})\hspace{6.67em}-w\left(t)\left(a\left(x){\left(-\Delta )}^{m}h,{\left(-\Delta )}^{k}{v}_{n,2}).\end{array}Then applying I−PnI-{P}_{n}to the first equation of (3.8), we obtain (4.72)vn,2=dun,2dt+εun,2−hw(t).{v}_{n,2}=\frac{{\rm{d}}{u}_{n,2}}{{\rm{d}}t}+\varepsilon {u}_{n,2}-hw\left(t).Combining the processing method of Lemma 4.1, (4.73)ddt[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]+2(a(x)∇m+kvn,2,∇m+kvn,2)−6+ε4a00‖∇m+kvn,2‖2−3ε‖∇kvn,2‖2+2εb0M0(C(‖∇mu‖2))(a(x)∇m+kun,2,∇m+kun,2)−3εa00b0M04‖∇m+kun,2‖2−2ε2(a(x)∇m+kun,2,∇m+kun,2)−ε2a00‖∇m+kun,2‖2+ε3‖∇kun,2‖2≤∑i=1m−k(1−αi)a004(1−αi)(m−k)1+αi1−αi(Cm−kiaiCi)21−αi‖vn,2‖2+2ε∑i=1m−k(1−αi)a008αi(m−k)αiαi−12(m−k)a00b0M0(Cm−kiaiCi)211−αi‖vn,2‖2+2(ε−1a00−1b0M0)11−αi∑i=1m−k(1−αi)a008αi(m−k)αiαi−1(Cm−kiaiCi(1+‖∇mu‖2q))21−αi‖vn,2‖2+4ε−1a00−1b0a02M0(1+‖∇mu‖2q)2∣w(t)∣2‖∇m+kh‖2+2λn+1−ma00‖∇kf(x,t)‖2+2Ck1(r01p(τ,w)+‖ϕ5(x)‖2)+5a00−1a02∣w(t)∣2‖∇m+kh‖2+2ε∣w(t)∣2‖∇kh‖2+4a00−1∣w(t)∣2∑i=1m−k(Cm−kiai)2λ1−i‖∇m+kh‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}+2\left(a\left(x){\nabla }^{m+k}{v}_{n,2},{\nabla }^{m+k}{v}_{n,2})-\frac{6+\varepsilon }{4}{a}_{00}\Vert {\nabla }^{m+k}{v}_{n,2}{\Vert }^{2}-3\varepsilon \Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}+2\varepsilon {b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})-\frac{3\varepsilon {a}_{00}{b}_{0}{M}_{0}}{4}\Vert {\nabla }^{m+k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}-2{\varepsilon }^{2}\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})-{\varepsilon }^{2}{a}_{00}\Vert {\nabla }^{m+k}{u}_{n,2}{\Vert }^{2}+{\varepsilon }^{3}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}\le \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{4\left(1-{\alpha }_{i})\left(m-k)}\right)}^{\tfrac{1+{\alpha }_{i}}{1-{\alpha }_{i}}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+2\varepsilon \mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left(\frac{2\left(m-k)}{{a}_{00}{b}_{0}{M}_{0}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i})}^{2}\right)}^{\tfrac{1}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+\hspace{0.25em}2{\left({\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{M}_{0})}^{\tfrac{1}{1-{\alpha }_{i}}}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}\left(1-{\alpha }_{i}){\left(\frac{{a}_{00}}{8{\alpha }_{i}\left(m-k)}\right)}^{\tfrac{{\alpha }_{i}}{{\alpha }_{i}-1}}{\left({C}_{m-k}^{i}{a}_{i}{C}_{i}\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q}))}^{\tfrac{2}{1-{\alpha }_{i}}}\Vert {v}_{n,2}{\Vert }^{2}\\ \hspace{2.0em}+4{\varepsilon }^{-1}{a}_{00}^{-1}{b}_{0}{a}_{0}^{2}{M}_{0}{\left(1+\Vert {\nabla }^{m}u{\Vert }^{2q})}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+\frac{2{\lambda }_{n+1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}\\ \hspace{2.0em}+2{C}_{k1}\left({r}_{01}^{p}\left(\tau ,w)+\Vert {\phi }_{5}\left(x){\Vert }^{2})+5{a}_{00}^{-1}{a}_{0}^{2}| w\left(t){| }^{2}\Vert {\nabla }^{m+k}h{\Vert }^{2}+2\varepsilon | w\left(t){| }^{2}\Vert {\nabla }^{k}h{\Vert }^{2}\\ \hspace{2.0em}+4{a}_{00}^{-1}| w\left(t){| }^{2}\mathop{\displaystyle \sum }\limits_{i=1}^{m-k}{\left({C}_{m-k}^{i}{a}_{i})}^{2}{\lambda }_{1}^{-i}\Vert {\nabla }^{m+k}h{\Vert }^{2}.\end{array}Then there is a positive constant σ1{\sigma }_{1}(4.73) can be reduced to (4.74)ddt[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]+σ1[‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2]≤Ck4(1+∣w(t)∣2)+2λn+1−ma00‖∇kf(x,t)‖2.\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}+{\sigma }_{1}\left[\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}]\\ \hspace{1.0em}\le {C}_{k4}\left(1+| w\left(t){| }^{2})+\frac{2{\lambda }_{n+1}^{-m}}{{a}_{00}}\Vert {\nabla }^{k}f\left(x,t){\Vert }^{2}.\end{array}when k=mk=m, (4.74) also holds.□Integrating (4.74) over (τ−t,τ)\left(\tau -t,\tau )with t≥0t\ge 0, we get for all n≥Nk1n\ge {N}_{k1}(4.75)‖∇kvn,2‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2,∇m+kun,2)+ε2‖∇kun,2‖2≤e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk4(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}{v}_{n,2}{\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2},{\nabla }^{m+k}{u}_{n,2})+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}{\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k4}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}d\xi .\end{array}Replacing wwby θ−τw{\theta }_{-\tau }win (4.75), we obtain for every t∈R+,τ∈R,w∈Ω1,n≥Nk1t\in {R}^{+},\tau \in R,w\in {\Omega }_{1},n\ge {N}_{k1}(4.76)‖∇kvn,2(τ,τ−t,θ−τw)‖2+(b0M0(C(‖∇mu‖2))−ε)×(a(x)∇m+kun,2(τ,τ−t,θ−τw),∇m+kun,2(τ,τ−t,θ−τw))+ε2‖∇kun,2(τ,τ−t,θ−τw)‖2≤e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]+e−σ1τ∫τ−tτeσ1ξCk4(1+∣w(ξ)∣2)dξ+2λ1−ma00e−σ1τ∫τ−tτeσ1ξ‖∇kf(x,ξ)‖2dξ.\begin{array}{l}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\\ \hspace{1.0em}\times \hspace{0.33em}\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w),{\nabla }^{m+k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w))+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau ,\tau -t,{\theta }_{-\tau }w){\Vert }^{2}\\ \hspace{1.0em}\le {e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))\\ \hspace{2.0em}+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]+{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }{C}_{k4}\left(1+| w\left(\xi ){| }^{2}){\rm{d}}\xi +\frac{2{\lambda }_{1}^{-m}}{{a}_{00}}{e}^{-{\sigma }_{1}\tau }\underset{\tau -t}{\overset{\tau }{\displaystyle \int }}{e}^{{\sigma }_{1}\xi }\Vert {\nabla }^{k}f\left(x,\xi ){\Vert }^{2}{\rm{d}}\xi .\end{array}From (3.8), h∈Vm+kh\in {V}_{m+k}, f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1})Lemma 4.1 and (uτ−t,vτ−t)∈Bk(τ−t,θ−τw)\left({u}_{\tau -t},{v}_{\tau -t})\in {B}_{k}\left(\tau -t,{\theta }_{-\tau }w)for t→+∞t\to +\infty (4.77)e−σ1t[‖∇kvn,2(τ−t)‖2+(b0M0(C(‖∇mu‖2))−ε)(a(x)∇m+kun,2(τ−t),∇m+kun,2(τ−t))+ε2‖∇kun,2(τ−t)‖2]→0.{e}^{-{\sigma }_{1}t}{[}\Vert {\nabla }^{k}{v}_{n,2}\left(\tau -t){\Vert }^{2}+\left({b}_{0}{M}_{0}\left(C\left(\Vert {\nabla }^{m}u{\Vert }^{2}))-\varepsilon )\left(a\left(x){\nabla }^{m+k}{u}_{n,2}\left(\tau -t),{\nabla }^{m+k}{u}_{n,2}\left(\tau -t))+{\varepsilon }^{2}\Vert {\nabla }^{k}{u}_{n,2}\left(\tau -t){\Vert }^{2}]\to 0.Combining (4.69), (4.70), (4.76), (4.77), and Lemma 4.1, we can get the conclusion of Lemma 4.2.Lemma 4.2 is proved.5The existence of the family of random attractorsIn this section, we shall prove the existence of the family of random pullback attractors for system (3.8). From Lemma 4.1, we know that for P−a.e.Dk={Dk(τ,w):τ∈R,w∈Ω1}∈DkP-{a.e.}\hspace{0.25em}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}and w∈Ω1w\in {\Omega }_{1}, there exists Tk=Tk(Dk,w){T}_{k}={T}_{k}\left({D}_{k},w)such that for all t≥Tkt\ge {T}_{k}(5.1)‖v(τ,τ−t,θ−τw,vτ−t)‖Vk2+‖∇m+ku(τ,τ−t,θ−τw,uτ−t)‖Vm+k2≤r1k(τ,w).\Vert v\left(\tau ,\tau -t,{\theta }_{-\tau }w,{v}_{\tau -t}){\Vert }_{{V}_{k}}^{2}+\Vert {\nabla }^{m+k}u\left(\tau ,\tau -t,{\theta }_{-\tau }w,{u}_{\tau -t}){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w).Let (5.2)Bk(τ,w)={(u,v)∈Vm+k×Vk:‖v‖Vk2+‖u‖Vm+k2≤r1k(τ,w)}.{B}_{k}\left(\tau ,w)=\left\{\left(u,v)\in {V}_{m+k}\times {V}_{k}:\Vert v{\Vert }_{{V}_{k}}^{2}+\Vert u{\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w)\right\}.Then, by (5.2), Bk={Bk(τ,w)}w∈Ω1{B}_{k}={\left\{{B}_{k}\left(\tau ,w)\right\}}_{w\in {\Omega }_{1}}are the closed absorption sets of Φk{\Phi }_{k}in Xk{X}_{k}. We are now ready to prove the asymptotic compactness of Φk{\Phi }_{k}in Xk{X}_{k}.Lemma 5.1Suppose MMsatisfies (M),h(x)∈Vm+k(Ω)\left(M),h\left(x)\in {V}_{m+k}\left(\Omega ), (3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies, (F1)\left({F}_{1}), then Φk{\Phi }_{k}is asymptotically compact in Xk{X}_{k}, that is, for every τ∈R,w∈Ω1\tau \in R,w\in {\Omega }_{1}, the sequence {Φk(ti,τ−ti,θ−tiw,(uτ,i,vτ,i))}\left\{{\Phi }_{k}\left({t}_{i},\tau -{t}_{i},{\theta }_{-{t}_{i}}w,\left({u}_{\tau ,i},{v}_{\tau ,i}))\right\}has a convergent subsequence in Xk{X}_{k}provided ti→∞{t}_{i}\to \infty and(uτ,i,vτ,i)∈Dk(τ−ti,θ−tiw);Dk={Dk(τ,w):τ∈R,w∈Ω1}∈Dk.\left({u}_{\tau ,i},{v}_{\tau ,i})\in {D}_{k}\left(\tau -{t}_{i},{\theta }_{-{t}_{i}}w);\hspace{0.33em}{D}_{k}=\left\{{D}_{k}\left(\tau ,w):\tau \in R,w\in {\Omega }_{1}\right\}\in {{\mathcal{D}}}_{k}.ProofWe first let ti→∞{t}_{i}\to \infty , it follows from Lemma 4.1 that there exist i1=i1(τ,w,Dk)>0{i}_{1}={i}_{1}\left(\tau ,w,{D}_{k})\gt 0such that for every i≥i1i\ge {i}_{1}(5.3)‖v(τ,τ−ti,θ−τw,vτ)‖Vk2+‖u(τ,τ−ti,θ−τw,uτ)‖Vm+k2≤r1k(τ,w),\Vert v\left(\tau ,\tau -{t}_{i},{\theta }_{-\tau }w,{v}_{\tau }){\Vert }_{{V}_{k}}^{2}+\Vert u\left(\tau ,\tau -{t}_{i},{\theta }_{-\tau }w,{u}_{\tau }){\Vert }_{{V}_{m+k}}^{2}\le {r}_{1k}\left(\tau ,w),next by using Lemma 4.2 for ∀ηk>0\forall \hspace{-0.25em}{\eta }_{k}\gt 0, there are ik2=ik2(ηk,w,Bk){i}_{k2}={i}_{k2}\left({\eta }_{k},w,{B}_{k})and Nk=Nk(ηk,w)>0{N}_{k}={N}_{k}\left({\eta }_{k},w)\gt 0such that for every ik≥ik2{i}_{k}\ge {i}_{k2}(5.4)‖(I−Pn)v(τ,τ−tki,θ−τw,vτ)‖Vk2+‖(I−Pn)u(τ,τ−tki,θ−τw,uτ)‖Vm+k2≤ηk.\Vert \left(I-{P}_{n})v\left(\tau ,\tau -{t}_{ki},{\theta }_{-\tau }w,{v}_{\tau }){\Vert }_{{V}_{k}}^{2}+\Vert \left(I-{P}_{n})u\left(\tau ,\tau -{t}_{ki},{\theta }_{-\tau }w,{u}_{\tau }){\Vert }_{{V}_{m+k}}^{2}\le {\eta }_{k}.By using (5.3), we find that {PNk(u(τ,τ−ti,w),v(τ,τ−ti,w))}\left\{{P}_{{N}_{k}}\left(u\left(\tau ,\tau -{t}_{i},w),v\left(\tau ,\tau -{t}_{i},w))\right\}is bounded in PNkXk{P}_{{N}_{k}}{X}_{k}and PNkXk{P}_{{N}_{k}}{X}_{k}is finite dimensional, which associates with (5.4) implies that {(u(τ,τ−ti,w),v(τ,τ−ti,w))}\left\{\left(u\left(\tau ,\tau -{t}_{i},w),v\left(\tau ,\tau -{t}_{i},w))\right\}is precompact in Xk{X}_{k}.□Theorem 5.2Suppose MMsatisfies (M),h(x)∈Vm+k(Ω)\left(M),\hspace{0.33em}h\left(x)\in {V}_{m+k}\left(\Omega ), (3.2)–(3.6) hold f(x,t)f\left(x,t)satisfies (F1)\left({F}_{1}), then the family of cocycles Φk{\Phi }_{k}generated by (3.8) has a family of pullback Dk{{\mathcal{D}}}_{k}attractors {Ak}={{Ak(τ,w)}∈Dk(k=1,2,…,m)}\left\{{A}_{k}\right\}=\left\{\left\{{A}_{k}\left(\tau ,w)\right\}\in {{\mathcal{D}}}_{k}\hspace{0.25em}\left(k=1,2,\ldots ,m)\right\}in Xk{X}_{k}and can be expressed as follows: Ak(τ,w)=⋂τ≥0⋃t≥τΦk(t,τ−t,θ−tw,Bk(τ−t,θ−tw))¯,τ∈R,w∈Ω1.{A}_{k}\left(\tau ,w)=\bigcap _{\tau \ge 0}\overline{\bigcup _{t\ge \tau }{\Phi }_{k}\left(t,\tau -t,{\theta }_{-t}w,{B}_{k}\left(\tau -t,{\theta }_{-t}w))},\hspace{1em}\tau \in R,\hspace{1em}w\in {\Omega }_{1}.ProofFrom (5.2), Lemmas 5.1 and 2.10, the conclusion of Theorem 5.2 can be obtained.Theorem 5.2 is proved.□Note 5.3Theorem 5.2 shows the family of cocycles Φk{\Phi }_{k}generated by (3.8) has a unique pullback attractor Ak{A}_{k}, respectively, in the space Xk(k=0,1,…,m){X}_{k}\left(k=0,1,\ldots ,m), which together form a family of pullback attractors {Ak}\left\{{A}_{k}\right\}. At the same time, according to Lemma 4.1 and (5.2) and the tight embedding of Xk↪X0,k=1,2,…,m{X}_{k}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{X}_{0},k=1,2,\ldots ,m, get the corresponding a family of pullback attractors {Ak}\left\{{A}_{k}\right\}, which is (Xk,X0)\left({X}_{k},{X}_{0})the family of random weak attractors, which means that the family of cocycles Φk{\Phi }_{k}has uniformly asymptotically compact absorption sets Bk(τ,w)⊂X0,k=1,2,…,m{B}_{k}\left(\tau ,w)\subset {X}_{0},k=1,2,\ldots ,m, where Bk(τ,w){B}_{k}\left(\tau ,w)are the bounded sets in Xk{X}_{k}, i.e., Φk{\Phi }_{k}are asymptotically compact in X0{X}_{0}.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: nonautonomous higher-order Kirchhoff equation; partial differential equations; variable coefficient; the family of random attractors; additive noise; 37B55; 35B41; 35G31; 60H15

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