Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result

Abstract

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies $g'(t)\leq -\xi (t)g^{r}(t)$ g ′ ( t ) ≤ − ξ ( t ) g r ( t ) , $t\geq 0$ t ≥ 0 , $1\leq r< \frac{3}{2}$ 1 ≤ r < 3 2 . The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.