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Applied Mathematics & Optimization

, Volume 73, Issue 2, pp 251–269 | Cite as

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

  • Hassan Yassine
  • Ali Abbas
Article
  • 187 Downloads

Abstract

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation:
$$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$
in \({\mathbb {R}}^+\times \Omega \), with Dirichlet boundary conditions, where \(\Omega \) is a bounded domain in \({\mathbb {R}}^n\) 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.

Keywords

Evolutionary integral equation Semilinear Stabilization  Łojasiewicz–Simon inequality 

Mathematics Subject Classification

28C15 46E05 28C99 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesLebanese UniversityBaalbek-ZahleLebanon

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