A new transformation for the subcritical fast diffusion equation with source and applications

TL;DR

This paper introduces a novel transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source, enabling classification of self-similar solutions and analyzing blow-up behavior.

Contribution

A new transformation is developed that acts as a symmetry with respect to critical exponents, aiding in the classification of self-similar solutions for the fast diffusion equation with source.

Findings

Classified self-similar solutions with or without finite time blow-up.

Extended previous results using the new transformation.

Provided insights into the behavior of solutions near critical exponents.

Abstract

A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$ and with dimension and exponents $$ N\geq3, \quad 0<m<m_c:=\frac{N-2}{N}, \quad \sigma\in(-2,\infty), $$ is introduced. It plays a role of a kind of symmetry with respect to the critical exponents $$ m_s=\frac{N-2}{N+2}, \quad p_L(\sigma)=1+\frac{\sigma(1-m)}{2}, \quad p_s(\sigma)=\frac{m(N+2\sigma+2)}{N-2}. $$ This transformation is then applied for classifying self-similar solutions with or without finite time blow-up to the subcritical fast diffusion equation with source when $p>\max\{1,p_L(\sigma)\}$, having as starting point previous results by the authors.