Eternal solutions for a reaction-diffusion equation with weighted reaction

TL;DR

This paper establishes the existence, uniqueness, and classification of eternal self-similar solutions for a weighted reaction-diffusion equation, revealing conditions under which such solutions exist or do not.

Contribution

It provides the first rigorous proof of eternal solutions in self-similar form for this class of weighted reaction-diffusion equations, including their classification and non-existence results.

Findings

Existence and uniqueness of eternal solutions when m+p ≥ 2.

No eternal solutions exist when m+p < 2.

Classification of solutions with different interface behaviors for m+p > 2.

Abstract

We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$, with $m>1$, $0<p<1$ and the critical value for the weight $$ \sigma=\frac{2(1-p)}{m-1}. $$ Existence and uniqueness of some specific solution holds true when $m+p\geq2$. On the contrary, no eternal solution exists if $m+p<2$. We also classify exponential self-similar solutions with a different interface behavior when $m+p>2$. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.