Well-posedness and stability for a viscoelastic Petrovsky equation with a localized nonlinear damping

Abstract

In this paper, we consider a viscoelastic Petrovsky equation with localised nonlinear damping in bounded domain. The nonlinear damping is effective only in a neighborhood of a suitable subset of the boundary. Using the Faedo–Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, exponential decay results of the energy are established via suitable Lyapunov functionals