Electron. J. Diff. Eqns., Vol. 2002(2002), No. 44, pp. 1-14.

Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping

Marcelo M. Cavalcanti, Valeria N. Domingos Cavalcanti, & Juan A. Soriano

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.

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Marcelo M. Cavalcanti
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: marcelo@gauss.dma.uem.br
Valeria N. Domingos Cavalcanti
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: valeria@gauss.dma.uem.br
Juan Amadeo Soraino
Universidade Estadual de Maringa,
87020-900 Maringa-PR, Brazil
e-mail: soriano@gauss.dma.uem.br

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