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We perform a thorough study of the blow up profiles associated to thefollowing second order reaction-diffusion equation with non-homogeneousreaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ in the rangeof exponents $1<p<m$ and $\sigma>0$. We classify blow up solutions inself-similar form, that are likely to represent typical blow up patterns forgeneral solutions. We thus show that the non-homogeneous coefficient$|x|^{\sigma}$ has a strong influence on the qualitative aspects related to thefinite time blow up. More precisely, for $\sigma\sim0$, blow up profiles havesimilar 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{occursonly at space infinity}, in strong contrast with the homogeneous case. Thiswork is a part of a larger program of understanding the influence of unboundedweights on the blow up behavior for reaction-diffusion equations.

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