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We study the blow up profiles associated to the following second orderreaction-diffusion equation with non-homogeneous reaction: $$\partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u, $$ with $\sigma>0$. Throughthis study, we show that the non-homogeneous coefficient $|x|^{\sigma}$ has astrong influence on the blow up behavior of the solutions. First of all, itfollows that finite time blow up occurs for self-similar solutions $u$, afeature that does not appear in the well known autonomous case $\sigma=0$.Moreover, we show that there are three different types of blow up self-similarprofiles, depending on whether the exponent $\sigma$ is closer to zero or not.We also find an explicit blow up profile. The results show in particular that\emph{global blow up} occurs when $\sigma>0$ is sufficiently small, while for$\sigma>0$ sufficiently large blow up \emph{occurs only at infinity}, and wegive prototypes of these phenomena in form of self-similar solutions withprecise behavior. This work is a part of a larger program of understanding theinfluence of non-homogeneous weights on the blow up sets and rates.

Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction with linear growth