Threshold property of a singular stationary solution for semilinear heat equations with exponential growth
Kotaro Hisa, Yasuhito Miyamoto
TL;DR
The authors address the Cauchy problem for the semilinear heat equation $\partial_t u-\Delta u=f(u)$ on $\mathbb{R}^N$ ($N\ge3$) with nonlinearities of exponential or faster growth, constructing a positive radial singular stationary solution $u^*$ that blows up at the origin. They prove a sharp threshold phenomenon: if $0\le u_0\le u^*$ (with $u_0\not\equiv u^*$) there exists a global nonnegative solution, while if $u_0\ge u^*$ (with $u_0\not\equiv u^*$) there is no nonnegative local-in-time solution. The analysis combines a monotone iteration scheme to obtain extremal solutions, a weak-solution framework, and carefully designed comparisons with the singular stationary solution, extending the threshold paradigm known for power and pure exponential nonlinearities to a broader class of growth conditions. These results illuminate how the full initial-data shape relative to $u^*$ governs global existence versus immediate blow-up in this setting.
Abstract
Let $N\ge 3$. We are concerned with a Cauchy problem of the semilinear heat equation \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] 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 $u_0\not\equiv u^*$.