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

TL;DR

This paper investigates how a weighted reaction term affects blow-up behavior in a quasilinear reaction-diffusion equation, revealing new types of blow-up profiles and the influence of the weight exponent on blow-up location and nature.

Contribution

It identifies the existence of finite-time blow-up for self-similar solutions in a non-homogeneous setting and classifies different blow-up profiles based on the weight exponent.

Findings

Finite-time blow-up occurs for self-similar solutions when the weight exponent is positive.

Three types of blow-up profiles are identified depending on the value of the weight exponent.

Explicit blow-up profiles are constructed, illustrating global blow-up and blow-up at infinity.

Abstract

We study the blow up profiles associated to the following second order reaction-diffusion equation with non-homogeneous reaction: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u, $$ 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 \emph{global blow up} occurs when $\sigma>0$ is sufficiently small, while for $\sigma>0$ sufficiently large blow up \emph{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.