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

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 $m\to 1$.