Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

Abstract

We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight:

$$\begin{aligned} \partial _tu=\partial _{xx}(u^m) + |x|^{\sigma }u^p, \end{aligned}$$

posed for $x\in {\mathbb {R}}$, $t\ge 0$, where $m>1$, $0<p<1$ such that $m+p=2$ and $\sigma >2$ completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for $\sigma >2$. Moreover all the blow-up profiles have compact support and their supports are localized: there exists an explicit $\eta >0$ such that any blow-up profile satisfies $\mathrm{supp}\,f\subseteq [0,\eta ]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin.