Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipation

We consider the damped semilinear viscoelastic wave equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u'' - \Delta u + \int_0^t h (t - \tau ) div\{ a\nabla u(\tau )\} d\tau + g(u') = 0 in \Omega \times (0,\infty )$$ \end{document} with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.