On the Cauchy problem for a general fractional porous medium equation with variable density

TL;DR

This paper investigates the well-posedness of a fractional porous medium equation with variable density, establishing existence, and analyzing conditions for uniqueness and nonuniqueness based on the density's behavior at infinity.

Contribution

It introduces new results on existence and uniqueness for a fractional porous medium equation with variable density, considering different asymptotic behaviors.

Findings

Existence of weak energy solutions is proven.

Uniqueness depends on the density's behavior at infinity.

Nonuniqueness is also demonstrated under certain conditions.

Abstract

We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according with the behavior of the density at infinity.