General Decay Rate Estimate for the Energy of a Weak Viscoelastic Equation With an Internal Time-Varying Delay Term

In this paper we consider the weak viscoelastic equation with an internal time-varying delay term uₜₜ(x,t)-Δ u(x,t)+α(t)∫₀ᵗg(t-s)Δ u(x,s)\, ds +a₀uₜ(x,t)+a₁uₜ(x,t-τ(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 α and g.