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

Abstract

<p style='text-indent:20px;'>We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in any space dimension <inline-formula><tex-math id="M1">\begin{document}$ x\in \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula> and with exponents <inline-formula><tex-math id="M3">\begin{document}$ m>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p\in(0, 1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \sigma>2(1-p)/(m-1) $\end{document}</tex-math></inline-formula>. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are <i>compactly supported</i> and might present two different types of interface behavior and three different possible <i>good behaviors</i> near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>. This paper generalizes in dimension <inline-formula><tex-math id="M7">\begin{document}$ N>1 $\end{document}</tex-math></inline-formula> previous results by the authors in dimension <inline-formula><tex-math id="M8">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula> and also includes some finer classification of the profiles for <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> large that is new even in dimension <inline-formula><tex-math id="M10">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>