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 case n>2n>2, as time goes to infinity, of regular solutions for a nonlinear coupled system of wave equations with memory and weak dampingutt−Δu+∫0tg1(t−s)Δu(s)ds+αut+h(u−v)=0inQˆ,vtt−Δv+∫0tg2(t−s)Δv(s)ds+αvt−h(u−v)=0inQˆ, in a noncylindrical domains Qˆ of Rn+1(n⩾1)(n⩾1) under suitable hypotheses on the scalar functions h  , g1g1 and g2g2, 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⩾1n⩾1.