Fractional porous media equations: existence and uniqueness of weak solutions with measure data

TL;DR

This paper establishes the existence and uniqueness of weak solutions for fractional porous media equations driven by the fractional Laplacian, including cases with measure data and weighted variants, advancing the mathematical understanding of these nonlinear PDEs.

Contribution

It proves the existence and uniqueness of solutions with measure data for fractional porous media equations, including weighted cases with singularities, and addresses the initial trace problem.

Findings

Existence of solutions with measure data

Uniqueness of solutions including initial trace

Existence and uniqueness of Barenblatt-type solutions

Abstract

We prove existence and uniqueness of solutions to a class of porous media equations driven by the fractional Laplacian when the initial data are positive finite Radon measures on the Euclidean space. For given solutions without a prescribed initial condition, the problem of existence and uniqueness of the initial trace is also addressed. By the same methods we can also treat weighted fractional porous media equations, with a weight that can be singular at the origin, and must have a sufficiently slow decay at infinity (power-like). In particular, we show that the Barenblatt-type solutions exist and are unique. Such a result has a crucial role in [24], where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in [42]