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: $${\partial }_{t}u={\partial }_{xx}(u^m)+|x{|}^{\sigma }{u}^p,$$ posed for $x\in \Re$, $t\ge 0$, 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 \emph{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.

38 pages, 7 figures