A new transformation for the subcritical fast diffusion equation with source and applications
Razvan Gabriel Iagar, Ariel Sánchez
TL;DR
This work addresses the subcritical fast diffusion equation with a spatially inhomogeneous source $\partial_tu=\Delta u^m+|x|^\sigma u^p$ in $\mathbb{R}^N$ ($N\ge3$, $0<m<m_c$) by introducing a new radial self-map that connects solutions with different parameter sets via transformed exponents. The transformation reveals a symmetry with respect to the Sobolev exponent $m_s$ and critical exponents $p_L(\sigma)$ and $p_s(\sigma)$, enabling a unified analysis of self-similar solutions. Using this map, the authors classify forward and backward self-similar solutions in the subcritical regime for $p>\max\{1,p_L(\sigma)\}$, detailing existence, nonexistence, and asymptotic decay/growth regimes and mapping extinction to blow-up through the inversion property. The results extend previous work (e.g., IMS23b, IS25c, IS22c) by completing the self-similar classification in the challenging subcritical range, and they provide a versatile framework for future functional-analytic study of nonlinear diffusion with inhomogeneous sources.
Abstract
A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$ and with dimension and exponents $$ N\geq3, \quad 0<m<m_c:=\frac{N-2}{N}, \quad σ\in(-2,\infty), $$ is introduced. It plays a role of a kind of symmetry with respect to the critical exponents $$ m_s=\frac{N-2}{N+2}, \quad p_L(σ)=1+\frac{σ(1-m)}{2}, \quad p_s(σ)=\frac{m(N+2σ+2)}{N-2}. $$ 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 $p>\max\{1,p_L(σ)\}$, having as starting point previous results by the authors.