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arXiv · 1811.10330

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

Razvan Gabriel Iagar·Ariel Sánchez
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Abstract

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 $1 0$. 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.

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Razvan Gabriel Iagar, Ariel Sánchez. 2018-11-26. Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction. https://doi.org/10.1016/j.jde.2020.10.006

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