Long-time behavior of a semilinear wave equation with memory

Abstract

In this paper we study the long-time dynamics of the semilinear viscoelastic equation utt−Δutt−Δu+∫0∞μ(s)Δu(t−s)ds+f(u)=h,(Formula presented.) defined in a bounded domain of (Formula presented.) with Dirichlet boundary condition. The functions (Formula presented.) and (Formula presented.) represent forcing terms and the kernel function (Formula presented.) is assumed to decay exponentially. Then, by exploring only the dissipation given by the memory term, we establish the existence of a global attractor to the corresponding dynamical system.