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

Abstract

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 ) ≤ − ξ ( t ) g r ( t ) $g'(t)\leq -\xi (t)g^{r}(t)$ , t ≥ 0 $t\geq 0$ , 1 ≤ r < 3 2 $1\leq r< \frac{3}{2}$ . 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.

Keywords

• Wave equation
• Balakrishnan–Taylor damping
• Delay term
• Global existence
• Decay estimates
• Blow up