Blow-up of solutions of a nonlinear viscoelastic wave equation

Abstract

In this paper, we consider the nonlinear viscoelastic equation u t t − Δ u + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ − △ u t = ∣ u ∣ p − 2 u , in Ω × [ 0 , T ] , with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g , we prove that there are solutions with positive initial energy that blow up in finite time.