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

TL;DR

This paper proves that under specific conditions, solutions to a high-energy viscoelastic Kirchhoff wave equation with damping and nonlinear source do not exist globally, highlighting limitations in solution behavior.

Contribution

It establishes the first global nonexistence results for high-energy solutions of the viscoelastic Kirchhoff wave equation with damping.

Findings

Solutions with high energy do not exist globally under certain conditions.

The nonexistence holds despite the presence of damping and nonlinear effects.

The results depend on assumptions on the kernel function g and initial data.

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.