Abstract
We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction:∂tu=∂xx(um)+|x|σup, in the range of exponents 1<p<m and σ>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|σ has a strong influence on the qualitative aspects related to the finite time blow up. More precisely, for σ∼0, blow up profiles have similar behavior to the well-established profiles for the homogeneous case σ=0, and typically global blow up occurs, while for σ>0 sufficiently large, there exist blow up profiles for which blow up 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.
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