A general fractional porous medium equation

TL;DR

This paper develops a comprehensive theory for the existence, uniqueness, and qualitative behavior of solutions to a fractional porous medium equation with various parameters, extending classical results to fractional diffusion contexts.

Contribution

It introduces a unified framework for fractional porous medium equations, establishing existence, uniqueness, and properties of solutions for a wide range of parameters and boundary conditions.

Findings

Existence and uniqueness of weak solutions for a broad parameter range.

Solutions form an $L^1$-contraction semigroup.

Different qualitative properties emerge depending on parameter regimes.

Abstract

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt] u(x,0) = f(x), & \qquad x\in\mathbb{R}^N.%. $$ We consider data $f\in L^1(\mathbb{R}^N)$ and all exponents $0<\sigma<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-\sigma)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<m\le m_*$ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above $m_*$. We also study the dependence of solutions on $f,m$ and $\sigma$. Moreover, we consider the above questions for the problem posed in a bounded domain.