Blow-up results for viscoelastic wave equations with weak damping

In this work we consider a viscoelastic wave equation of the form utt−Δu+∫0tg(t−s)Δu(s)ds+h(ut)=|u|p−2uwith Dirichlet boundary condition. There are much literature on the blow-up result of solutions for the wave equation with damping term having polynomial growth near zero. However, to my knowledge, there is no blow-up result of solutions for the viscoelastic wave equation without polynomial growth near zero assumption on the damping term. This work is devoted to study a finite time blow-up result of solution with nonpositive initial energy as well as positive initial energy without imposing any restrictive growth near zero assumption on the damping term.