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

Abstract

We consider the viscoelastic equation $$ u_{tt}(x,t)-M(\|\nabla u\|_2^2) \Delta u(x,t)+\int_0^t g(t-s)\Delta u(x,s)ds+u_t =|u|^{p-1}u $$ with suitable initial data and boundary conditions. Under certain assumptions on the kernel $g$ and the initial data, we establish a new blow-up result for arbitrary positive initial energy, by using simple analysis techniques.

Keywords

• Viscoelastic equation
• blow-up
• arbitrary positive initial energy