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

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...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Hisa, Kotaro, Miyamoto, Yasuhito
Format: Article
Language:English
Subjects:
Cauchy problems
Thermodynamics
Online Access:Get full text
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Summary: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 if \(u_0\ge u^*\) and \(u_0\not\equiv u^*\).
ISSN:2331-8422