  We study the separate variable blow-up patterns associated to the following
second order reaction-diffusion equation: $$ \partial_tu=\Delta u^m +
|x|^{\sigma}u^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$,
dimension $N\geq2$ and $\sigma>0$. A new and explicit critical exponent $$
\sigma_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the
blow-up profiles is given. The most interesting contribution of the paper is
showing that existence and behavior of the blow-up patterns is split into
different regimes by the critical exponent $\sigma_c$ and also depends strongly
on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend
previous works of the authors in dimension $N=1$.
