Global nonexistence of solutions for the viscoelastic wave equation of Kirchhoff type with high energy

Abstract

In this paper we consider the viscoelastic wave equation of Kirchhoff type: $$ u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int_{0}^{t}g(t-s)\Delta u(s){\rm d}s+u_{t}=|u|^{p-1}u $$ with Dirichlet boundary conditions. Under some suitable assumptions on $g$ and the initial data, we established a global nonexistence result for certain solutions with arbitrarily high energy.