Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction

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 $N=1$.