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

Abstract

We perform a thorough study of the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $${\partial }_{t}u={\partial }_{xx}(u^m)+|x{|}^{\sigma }{u}^p,$$ 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 $\sigma \sim 0$, 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.

References in corpus (1)

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

Cited by in corpus (3)

• Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction
• Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential
• Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction