Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution equation with Neumann boundary conditions

Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution... The purpose of this paper is to investigate the time behavior of the solution of a weighted p-Laplacian evolution equation, given by $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t} = \text {div} \left( \gamma |\nabla u|^{p-2}\nabla u \right) &{} \text {on } (0,\infty )\times S, \\ \gamma |\nabla u|^{p-2}\nabla u\cdot \eta =0 &{} \text {on } (0,\infty )\times \partial S, \\ u(0,\cdot )=u_{0} &{} \text {on } S,\end{array}\right. } \end{aligned}$$ u t = div γ | ∇ u | p - 2 ∇ u on ( 0 , ∞ ) × S , γ | ∇ u | p - 2 ∇ u · η = 0 on ( 0 , ∞ ) × ∂ S , u ( 0 , · ) = u 0 on S , where $$n \in \mathbb {N}{\setminus } \{1\}$$ n ∈ N \ { 1 } , $$p \in (1,\infty ){\setminus } \{2\}$$ p ∈ ( 1 , ∞ ) \ { 2 } , $$S\subseteq \mathbb {R}^{n}$$ S ⊆ R n is an open, bounded and connected set of class $$C^{1}$$ C 1 , $$\eta $$ η is the unit outer normal on $$\partial S$$ ∂ S , and $$\gamma :S\rightarrow (0,\infty )$$ γ : S → ( 0 , ∞ ) is a bounded function which can be extended to an $$A_{p}$$ A p -Muckenhoupt weight on $$\mathbb {R}^{n}$$ R n . It will be proven that the solution of (0.1) converges in $$L^{1}(S)$$ L 1 ( S ) to the average of the initial value $$u_{0} \in L^{1}(S)$$ u 0 ∈ L 1 ( S ) . Moreover, a conservation of mass principle, an extinction principle and a decay rate for the solution will be derived. Nonlinear Differential Equations and Applications NoDEA Springer Journals

Asymptotic results for solutions of a weighted $${\varvec{p}}$$ p -Laplacian evolution equation with Neumann boundary conditions

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Mathematics; Analysis
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