Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary

In this article we study the degenerate System (rho(1), rho(2) greater than or equal to 0) subject to memory conditions on the boundary given by rho(1)(x)u(tt)-Deltau+alpha(u-v)=0 in Omegax]0,+infinity[, rho(2)(x)v(tt) - Deltav - alpha(u - v) 0 in Omega x]0, +infinity[, u = 0 on Gamma(0), u + integral(0)(t) g(1) (t - s) partial derivativeu/partial derivativenu(s) ds = 0 on Gamma(1) x]0, +infinity[, v = 0 on Gamma(0), v + integral(0)(t) g(2)(t - s) partial derivativenu/partial derivativenu (s) ds = 0 on Gamma(1) x]0, +infinity[, (u(0),v(0))=(u(0),v(0)) (rootrho(1)u(t)(0),rootrho(2)v(t)(0))=(rootrho(1)u(1),root(2)v(1)) in Omega, where Omega is a bounded region in R-n whose boundary is partitioned into disjoint sets Gamma(0), Gamma(1). We prove that the dissipations given by the memory terms are strong enough to guarantee exponential (or polynomial) decay provided the relaxation functions also decay exponentially (or polynomially) and with the same rate of decay. (C) 2003 Elsevier Inc. All rights reserved.