GENERAL DECAY RATE ESTIMATE FOR THE ENERGY OF A WEAK VISCOELASTIC EQUATION WITH AN INTERNAL TIME-VARYING DELAY TERM

Abstract

In this paper we consider the weak viscoelastic equation with an internal time-varying delay term $$u_{tt}(x,t)-\Delta u(x,t)+\alpha(t)\int_{0}^{t}g(t-s)\Delta u(x,s)\, {\rm d}s +a_{0}u_{t}(x,t)+a_{1}u_{t}(x,t-\tau(t))=0$$ in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay rate estimate for the energy, which depends on the behavior of both $\alpha$ and $g$.