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

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