Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term

Abstract

In this paper, we consider the weak viscoelastic wave equation utt−Δu+δΔut−σ(t)∫0tg(t−s)Δu(s)ds=|u|p−2u with dynamic boundary conditions, and nonlinear delay term. First, we prove a local existence theorem by using the Faedo–Galerkin approximations combined with a contraction mapping theorem. Secondly, we show that, under suitable conditions on the initial data and the relaxation function, the solution exists globally in time, in using the concept of stable sets. Finally, by exploiting the perturbed Lyapunov functionals, we extend and improve the previous result from Gerbi and Said-Houari (2011).