Asymptotic behaviour for wave equations with memory in a noncylindrical domains

Jorge Ferreira; Mauro De Lima Santos

doi:10.3934/cpaa.2003.2.511

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2003, Volume 2, Issue 4: 511-520. Doi: 10.3934/cpaa.2003.2.511

This issue Previous Article On the compactness of the stable set for rate independent processes Next Article On the dynamics of a mixed parabolic-gradient system

Asymptotic behaviour for wave equations with memory in a noncylindrical domains

Jorge Ferreira^1, and
Mauro De Lima Santos^2,

1.
Departamento de Matemática-DMA, Universidade Estadual de Maringá-UEM, Campus Universitário, Av. Colombo, 5790-Zona 7, CEP 87020-900, Maringá-Pr.
2.
Departamento de Matemática, Universidade Federal do Pará, Campus Universitário do Guamá, Rua Augusto Corrêa 01, Cep 66075-110, Pará

Received: February 28, 2003

Revised: June 30, 2003

Published: November 2003

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Abstract

In this paper we prove the exponential decay as time goes to infinity of regular solutions of the problem for the wave equations with memory and weak damping

$u_{t t}-\Delta u+\int^t_0g(t-s)\Delta u(s)ds + \alpha u_{t}=0$ in $\hat Q$

where $\hat Q$ is a non cylindrical domains of $\mathbb R^{n+1}$ $(n\ge1)$ with the lateral boundary $\hat{\sum}$ and $\alpha$ is a positive constant.

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Mathematics Subject Classification: 35K55, 35F30, 34B15.

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