We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:
$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $
posed in any space dimension $ x\in \mathbb{R}^N $, $ t\geq0 $ and with exponents $ m>1 $, $ p\in(0, 1) $ and $ \sigma>2(1-p)/(m-1) $. 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 $ \sigma $. This paper generalizes in dimension $ N>1 $ previous results by the authors in dimension $ N = 1 $ and also includes some finer classification of the profiles for $ \sigma $ large that is new even in dimension $ N = 1 $.
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