Energy Decay of the Solution for a Weak Viscoelastic Equation with a Time-Varying Delay

Abstract

In this paper, we consider a weak viscoelastic equation with internal time-varying delay $$\begin{aligned} u_{tt}(x, t)-\triangle u(x, t)+\alpha(t) \int_{0}^{t}g(t-s)\triangle u(x, s)ds+\mu u_{t}\bigl(x, t-\tau(t)\bigr)=0 \end{aligned}$$ in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay result for the energy. This work generalizes and improves earlier results in the literature.