Classification of blow-up and global existence of solutions to an initial $\textrm{Neumann}$ problem

TL;DR

This paper classifies when solutions to a nonlinear diffusion equation with Neumann boundary conditions blow up or exist globally, based on initial data, using potential well methods and differential inequalities.

Contribution

It provides a complete classification of solution behaviors, including thresholds for blow-up and global existence, and analyzes decay rates and asymptotic behavior.

Findings

Threshold results for global existence and blow-up based on initial energy

Decay rate of the L^2 norm for global solutions

Conditions for blow-up and global solutions with supercritical initial energy

Abstract

The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation driven by the $p(x)$-\hbox{Laplace} operator. Complete classification of global existence and blow-up in finite time of solutions is given when the initial data satisfies different conditions. Roughly speaking, we obtain a threshold result for the solution to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. Further, the decay rate of the $L^2$ norm is also obtained for global solutions. Sufficient conditions for the existence of global and blow-up solutions are also provided for supercritical initial energy. At last, we give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. This is a continuation of our previous work \cite{GG}.