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: $\partial_tu=\partial_{xx}(u^m) + |x|^σu,$ with $σ>0$. Through this study, we show that the non-homogeneous coefficient $|x|^σ$ 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 $σ=0$. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent $σ$ 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 $σ>0$ is sufficiently small, while for $σ>0$ 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.

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