Threshold property of a singular stationary solution for semilinear heat equations with exponential growth

TL;DR

This paper investigates a threshold phenomenon for solutions to a semilinear heat equation with exponential growth, identifying a singular stationary solution that determines global existence or blow-up based on initial data.

Contribution

It establishes the existence of a singular stationary solution and characterizes the initial data conditions for global existence versus blow-up.

Findings

Existence of a positive singular stationary solution $u^*$.

Global solutions exist if initial data is below $u^*$.

No solutions exist if initial data exceeds $u^*$.

Abstract

Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $f(0)=0$, $f$ is nonnegative, increasing and convex, $\log f(u)$ is convex for large $u>0$ and some additional assumptions are assumed. We establish a positive radial singular stationary solution $u^*$ such that $u^*(x)\to\infty$ as $|x|\to 0$. Then, we prove the following: The problem has a nonnegative global-in-time solution if $0\le u_0\le u^*$ and $u_0\not\equiv u^*$, while the problem has no nonnegative local-in-time solutions $u$ such that $u\ge u^*$ if $u_0\ge u^*$ and $u_0\not\equiv u^*$.