Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

TL;DR

This paper establishes the existence and uniqueness of anomalous exponential self-similar solutions for a weighted fast diffusion equation with reaction, revealing qualitative differences from classical solutions and highlighting a perfect equilibrium between diffusion and reaction effects.

Contribution

It introduces a new branch of anomalous eternal solutions for the subcritical fast diffusion equation with weighted reaction, including their existence, uniqueness, and qualitative behavior.

Findings

Solutions do not exhibit finite time extinction or blow-up.

Sign change in self-similar exponents at a critical parameter value.

Reaction term creates a perfect equilibrium between diffusion and reaction.

Abstract

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 0<m<m_c=\frac{N-2}{N}, \qquad p>1, $$ and the critical value for the weight $$ \sigma=\frac{2(p-1)}{1-m}. $$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a \emph{change of sign of both self-similar exponents} at $m=m_s=(N-2)/(N+2)$, leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.