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

We study the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $${\mathrm{\partial }}_{t}u={\mathrm{\partial }}_{xx}(u^m)+\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }u,$$ with $\sigma >0$. Through this study, we show that the non-homogeneous coefficient $\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }$ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions $u$, a feature 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-similar profiles, 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 we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates.