Global existence and energy decay of solutions to a viscoelastic wave equation with a delay term in the non-linear internal feedback

Abstract

We consider the viscoelastic wave equation in a bounded domain with a delay term in the non–linear internal feedback utt(x,t)−Δxu(x,t)+∫t0h(t−s)Δxu(x,s)ds+μ1g1(ut(x,t))+μ2g2(ut(x,t−τ))=0 and prove a global existence result using the energy method combined with the Faedo–Galerkin procedure under assumption of a relation between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we study the asymptotic behaviour of solutions using a perturbed energy method.