Energy decay of nonlocal viscoelastic equations with nonlinear damping and polynomial nonlinearity

Abstract

This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential decays.