Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 44, pp. 1-14.

Title: Exponential decay for the solution of semilinear viscoelastic wave
       equations with localized damping
   
Authors: Marcelo M. Cavalcanti (Univ. Estadual de Maringa, Brazil)
         Valeria N. Domingos Cavalcanti (Univ. Estadual de Maringa, Brazil)
         Juan A. Soriano (Univ. Estadual de Maringa, Brazil)

Abstract: 
 In this paper we obtain an exponential rate of decay for
 the solution of the viscoelastic nonlinear wave equation
 $$
 u_{tt}-\Delta u+f(x,t,u)+\int_0^tg(t-\tau )\Delta u(
 \tau )\,d\tau +a(x)u_t=0\quad \hbox{in }\Omega\times (0,\infty ).
 $$
 Here the damping term $a(x)u_t$ may be null for some part of 
 the domain $\Omega$. By assuming that the kernel $g$ in the
 memory term  decays exponentially, the damping effect allows
 us to avoid compactness arguments and and to reduce number of
 the energy estimates considered in the prior literature. We
 construct a suitable Liapunov functional and make use of
 the perturbed energy method.

Submitted March 07, 2002. Published May 22, 2002.
Math Subject Classifications: 35L05, 35L70, 35B40, 74D10.
Key Words: semilinear wave equation; memory; localized damping.