Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy

TL;DR

This paper investigates the long-term behavior of solutions to Kirchhoff-type equations, establishing thresholds for global existence or finite-time blow-up based on initial energy levels, and analyzing decay rates.

Contribution

It introduces a threshold criterion for solution behavior based on initial energy, extending understanding of Kirchhoff equations with arbitrary initial conditions.

Findings

Global solutions exist for subcritical initial energy.

Finite-time blow-up occurs for critical initial energy.

Decay rates of solutions are characterized in the global case.

Abstract

In this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of Kirchhoff type. Global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy. To be a little more precise, we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. The decay rate of the $L^2(\Omega)$ norm is also obtained for global solutions in these cases. Moreover, some sufficient conditions for the existence of global and blow-up solutions are also derived when the initial energy is supercritical.