On the existence and nonexistence of global solutions of the semilinear heat equation

TL;DR

This paper develops a new criterion using a similarity transform to determine whether solutions to the semilinear heat equation blow up or exist globally, extending previous results to a broader range of exponents.

Contribution

It introduces a novel approach employing a similarity transform and weighted Sobolev spaces to analyze solution behavior for a wider exponent range.

Findings

Established a new criterion for blow-up versus global existence.

Extended previous results from critical to subcritical Sobolev exponents.

Applied the potential well method in a transformed setting.

Abstract

We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. (\textit{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, \textbf{27} (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2/n$ is the Fujita exponent and $p_S=(n+2)/(n-2)$ (for $n\ge3$) is the critical Sobolev exponent.