The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain
Abstract
Description
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 $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, 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.
24 pages, replaces and improves previous version
24 pages, replaces and improves previous version