Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:\begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document}posed in any space dimension \begin{document}$ x\in \mathbb{R}^N $\end{document} , \begin{document}$ t\geq0 $\end{document} and with exponents \begin{document}$ m>1 $\end{document} , \begin{document}$ p\in(0, 1) $\end{document} and \begin{document}$ \sigma>2(1-p)/(m-1) $\end{document} . We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of \begin{document}$ \sigma $\end{document} . This paper generalizes in dimension \begin{document}$ N>1 $\end{document} previous results by the authors in dimension \begin{document}$ N = 1 $\end{document} and also includes some finer classification of the profiles for \begin{document}$ \sigma $\end{document} large that is new even in dimension \begin{document}$ N = 1 $\end{document} .