Asymptotic stability for a class of viscoelastic equations with general relaxation functions and the time delay

Abstract

The goal of the present paper is to study the viscoelastic wave equation with the time delay \[ |u_t|^\rho u_{tt}-\Delta u-\Delta u_{tt}+\int_0^tg(t-s)\Delta u(s)ds+\mu_1u_t(x,t)+\mu_2 u_t(x,t-\tau)=b|u|^{p-2}u\] under initial boundary value conditions, where $\rho,~b,~\mu_1$ are positive constants, $\mu_2$ is a real number, $\tau>0$ represents the time delay. By using the multiplier method together with some properties of the convex functions, the explicit and general stability results of energy are proved under the general assumption on the relaxation function $g$. This work generalizes and improves earlier results on the stability of the viscoelastic equations with the time delay in the literature.