Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction

TL;DR

This paper classifies self-similar blow-up profiles for a reaction-diffusion equation with a critically strong weighted reaction, revealing that such solutions exist with localized support and detailed behavior near the origin.

Contribution

It completes the analysis of a critical case for reaction-diffusion equations with weighted reactions, showing existence, support localization, and profile behavior.

Findings

Existence of self-similar blow-up solutions for >2

Profiles have compact, localized support

Detailed classification of profile behavior near the origin

Abstract

We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for $x\in\real$, $t\geq0$, 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 ${\rm supp}\,f\subseteq[0,\eta]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin.