The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain

TL;DR

This paper determines the critical exponent for the existence of global solutions to a semilinear heat equation with a quadratic potential or in an exterior domain, extending classical blow-up results.

Contribution

It calculates the Fujita-type critical exponent for semilinear heat equations with quadratic decay potentials and exterior domain conditions, including boundary cases.

Findings

Identifies the critical exponent p* for global existence versus blow-up.

Shows the equivalence of blow-up/global dichotomy in the potential case and exterior domain.

Provides conditions under which solutions blow up or exist globally.

Abstract

Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global solution dichotomy for \eqref{abstract} coincides with that for the corresponding problem in an exterior domain with the Dirichlet boundary condition, including the case in which $p$ is equal to the critical exponent.