  We study the blow up profiles associated to the following second order
reaction-diffusion equation with non-homogeneous reaction: $$
\partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u, $$ with $\sigma>0$. Through
this study, we show that the non-homogeneous coefficient $|x|^{\sigma}$ 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.
