Optimal decay rates of a nonlinear time-delayed viscoelastic wave equation

Abstract

This paper concerns a nonlinear viscoelastic wave equation with time-dependent delay. Under suitable relation between the weight of the delay and the weight of the term without delay, we prove the global existence of weak solutions by the combination of the Galerkin method and potential well theory. In addition, by assuming the minimal conditions on the $L^1(0,\infty)$ relaxation function $g$, namely, $g'(t)\leq-\xi(t)H(g(t))$, where $H$ is an increasing and convex function and $\xi$ is a nonincreasing differentiable function, and by using some properties of convex functions, we establish optimal explicit and general energy decay results. This result is new and substantially improves existing results in the literature.