Asymptotic behaviour for wave equations with memory in a noncylindrical domains

Abstract

In this paper we prove the exponential decay astime goes to infinity of regular solutions of the problem for thewave 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 apositive constant.