Anisotropic Fast Diffusion Equations
Abstract
We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite-mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.
Cite
@article{arxiv.2007.00122,
title = {Anisotropic Fast Diffusion Equations},
author = {Filomena Feo and Juan Luis Vázquez and Bruno Volzone},
journal= {arXiv preprint arXiv:2007.00122},
year = {2023}
}
Comments
62 pages. Typos corrected, references added, text provided with very detailed proofs. To appear in Nonlinear Analysis in 2023