  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.
