We study the blow up profiles associated to the following second order reaction-diffusion
equation with non-homogeneous reaction: disp-formula id="Equ56"mml:mtable mml:mtr
mml:mtd columnalign="right"partial derivative tu=partial derivative xx(um)+|x|sigma
u,mml:mtdmml:mtrmml:mtable 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 global blow up occurs when sigma 0
is sufficiently small, while for sigma >0 sufficiently large blow up 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.
