Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation

In this paper, we consider the nonlinear viscoelastic equationutt−Δu+∫0tg(t−τ)Δu(τ)dτ+ut|ut|m−2=u|u|p−2 with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g   and for p>mp>m, we prove that there are solutions with positive initial energy that blow up in finite time.