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

We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^σu^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $0<p<1$ such that $m+p=2$ and $σ>2$ 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 $σ>2$. Moreover all the blow-up profiles have compact support and their supports are \emph{localized}: there exists an explicit $η>0$ such that any blow-up profile satisfies ${\rm supp}\,f\subseteq[0,η]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin.