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

Abstract

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $${\partial }_{t}u=\Delta u^m+|x{|}^{\sigma }{u}^m,$$ posed for $x\in {\mathbb{R}}^N$, $t\ge 0$, where $m>1$, dimension $N\ge 2$ 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 $N\ge 4$ or $N\in \{2,3\}$. These results extend previous works of the authors in dimension $N=1$.