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

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 $p>1$, $n\ge 2$, $T\in (0,\mathrm{\infty }]$, $V(x)\sim \frac{\omega }{\mathrm{\mid }x{\mathrm{\mid }}^2}$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, for some $\omega \mathrm{\ne }0$, and $a(x)$ is on the order $\mathrm{\mid }x{\mathrm{\mid }}^m$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, for some $m\in (-\mathrm{\infty },\mathrm{\infty })$. A solution to the above equation is called global if $T=\mathrm{\infty }$. 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.