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.