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

Abstract

Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge 2$, $T\in (0,\infty ]$, $V(x)\sim \frac{\omega }{|x{|}^2}$ as $|x|\to \infty$, for some $\omega \ne 0$, 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 $p^{\ast }$ such that global solutions exist for $p>p^{\ast }$, while for $1<p\le p^{\ast }$, all solutions blow up in finite time. We also show that when $V\equiv 0$, 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.

24 pages, replaces and improves previous version