Global solvability and asymptotic behavior for a nonlinear coupled system of viscoelastic waves with memory in a noncylindrical domain

In this paper we prove the exponential decay in the casen>2, as time goes to infinity, of regular solutions for a nonlinear coupled system of wave equations with memory and weak damping utt − u + t 0 g1(t −s) u(s)ds +αut + h(u− v) =0 inQˆ , vtt − v + t 0 g2(t −s) v(s)ds +αvt −h(u −v) =0 inQˆ , in a noncylindrical domains Qˆ of n+1 (n 1) under suitable hypotheses on the scalar functions h, g1 and g2, and where α is a positive constant. We show that such dissipation is strong enough to produce uniform rate decay. Besides, the coupled is nonlinear which brings up some additional difficulties, which make the problem interesting. We establish existence and uniqueness of regular solutions for any n 1. © 2006 Elsevier Inc. All rights reserved