Blow up of solutions for viscoelastic wave equations of Kirchhoff type with arbitrary positive initial energy

Abstract

In this article we consider the nonlinear Viscoelastic wave equations of Kirchhoff type $$\displaylines{ u_{tt}-M( \| \nabla u\| ^2) \Delta u+\int_0^{t}g_1( t-\tau )\Delta u( \tau ) d\tau +u_t =( p+1)| v| ^{q+1}| u| ^{p-1}u, \cr v_{tt}-M( \| \nabla v\| ^2) \Delta v+\int_0^{t}g_2( t-\tau ) \Delta v( \tau ) d\tau +v_t=( q+1) | u| ^{p+1}| v| ^{q-1}v }$$ with initial conditions and Dirichlet boundary conditions. We proved the global nonexistence of solutions by applying a lemma by Levine, and the concavity method.

Keywords

• Blow up
• viscoelastic wave equation
• arbitrary positive initial energy