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

Abstract

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$|ut|?utt-Δutt-Δut-Δu+?0?k(s)Δu(t-s)ds+f(x,u)=g,??{t,?},in $${\mathbb {R}}^+\times \Omega $$R+×Ω, with Dirichlet boundary conditions, where $$\Omega $$Ω is a bounded domain in $${\mathbb {R}}^n$$Rn and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the ?ojasiewicz---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 ?ojasiewicz exponent and the decay of the term g.