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

We classify the self-similar blow-up profiles for 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 $$x\in {\mathbb {R}}$$ , $$t\ge 0$$ where $$m>1$$ $$02$$ completing analysis performed in a recent work this very interesting case was left aside. show finite time solutions form exist . Moreover all have compact support their supports are localized: there exists an explicit $$\eta >0$$ any profile satisfies $$\mathrm{supp}\,f\subseteq [0,\eta ]$$ This property is unexpected contrasting range $$m+p>2$$ also possible behaviors of near origin.