Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation

Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$ | u t | ρ u t t - Δ u t t - Δ u t - Δ u + ∫ 0 τ k ( s ) Δ u ( t - s ) d s + f ( x , u ) = g , τ ∈ { t , ∞ } , in $${\mathbb {R}}^+\times \Omega $$ R + × Ω , with Dirichlet boundary conditions, where $$\Omega $$ Ω is a bounded domain in $${\mathbb {R}}^n$$ R n and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the Łojasiewicz–Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the term g . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Long-Time Stabilization of Solutions to a Nonautonomous Semilinear Viscoelastic Equation

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Springer US
Copyright © 2016 by Springer Science+Business Media New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
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