Self-similar solutions to the time-fractional Porous-Medium Equation

TL;DR

This paper establishes the existence of self-similar solutions with finite mass for the time-fractional Porous-Medium Equation across all dimensions and exponents, characterizing their types and properties.

Contribution

It proves the existence of self-similar solutions for all relevant parameters, including the optimal range, and analyzes their behavior in different diffusion regimes.

Findings

Existence of self-similar solutions for all dimensions and exponents

Identification of solution types: compactly supported and heavy-tailed

Analysis of the linear case and the limit as m approaches 1

Abstract

We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of solution depending on the exponent: compactly supported solutions in the slow-diffusion range $m > 1$ and positive solutions with heavy tails in the sub-critical fast-diffusion range $m_c < m < 1$. The self-similar solutions in the linear case $m=1$ were already known explicitly obtained by the Fourier transform, and we discuss their properties in our settings and the limit $m \to 1$.