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

Abstract

A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $${\partial }_{t}u=\hat{I}”u^m+|x{|}^{\ddot{I}}ƒu^p,$$ posed for $(x,t)\in {\mathbb{R}}^N\times (0,\infty )$ and with dimension and exponents $$N\ge 3,0<m<m_c:=\frac{N-2}{N},\ddot{I}ƒ\in (-2,\infty ),$$ is introduced. It plays a role of a kind of symmetry with respect to the critical exponents $$m_s=\frac{N-2}{N+2},p_L(\ddot{I}ƒ)=1+\frac{\ddot{I}ƒ(1-m)}{2},p_s(\ddot{I}ƒ)=\frac{m(N+2\ddot{I}ƒ+2)}{N-2}.$$ 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 $p>\max \{1,p_L(\ddot{I}ƒ)\}$, having as starting point previous results by the authors.

arXiv admin note: substantial text overlap with arXiv:2307.04714