Electron. J. Diff. Eqns., Vol. 2007(2007), No. 128, pp. 1-18.

On the wave equations with memory in noncylindrical domains

Mauro L. Santos

In this paper we prove the exponential and polynomial decays rates in the case $n > 2$, as time approaches infinity of regular solutions of the wave equations with memory
 u_{tt}-\Delta u+\int^{t}_{0}g(t-s)\Delta u(s)ds=0 
 \quad \hbox{in } \widehat{Q}
where $\widehat{Q}$ is a non cylindrical domains of $\mathbb{R}^{n+1}$, $(n\geq 1)$. We show that the dissipation produced by memory effect is strong enough to produce exponential decay of solution provided the relaxation function $g$ also decays exponentially. When the relaxation function decay polynomially, we show that the solution decays polynomially with the same rate. For this we introduced a new multiplier that makes an important role in the obtaining of the exponential and polynomial decays of the energy of the system. Existence, uniqueness and regularity of solutions for any $n \ge 1$ are investigated. The obtained result extends known results from cylindrical to non-cylindrical domains.

Submitted March 8, 2007. Published October 2, 2007.
Math Subject Classifications: 35K55, 35F30, 34B15.
Key Words: Wave equation; noncylindrical domain; memory dissipation.

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Mauro de Lima Santos
Faculdade de Matemática, Universidade Federal do Pará
Campus Universitario do Guamá,
Rua Augusto Corrêa 01, Cep 66075-110, Pará, Brasil
email: ls@ufpa.br

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