  We perform a thorough study of the blow up profiles associated to the
following second order reaction-diffusion equation with non-homogeneous
reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ in the range
of exponents $1<p<m$ and $\sigma>0$. We classify blow up solutions in
self-similar form, that are likely to represent typical blow up patterns for
general solutions. We thus show that the non-homogeneous coefficient
$|x|^{\sigma}$ has a strong influence on the qualitative aspects related to the
finite time blow up. More precisely, for $\sigma\sim0$, blow up profiles have
similar behavior to the well-established profiles for the homogeneous case
$\sigma=0$, and typically \emph{global blow up} occurs, while for $\sigma>0$
sufficiently large, there exist blow up profiles for which blow up \emph{occurs
only at space infinity}, in strong contrast with the homogeneous case. This
work is a part of a larger program of understanding the influence of unbounded
weights on the blow up behavior for reaction-diffusion equations.
