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

Abstract

We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: ∂tu=∂xx(um)+|x|σu,with σ> 0. Through this study, we show that the non-homogeneous coefficient | x| σ 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 σ= 0. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent σ 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 σ> 0 is sufficiently small, while for σ> 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. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.