Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic 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 t | m − 2 = u | u | p − 2 with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g and for p > m , we prove that there are solutions with positive initial energy that blow up in finite time.