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

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 $${\mathbb R}^d$$ . 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 Grillo et al. (Discret Contin Dyn Syst 35:5927–5962, 2015), where the asymptotic behavior of solutions is investigated. Our uniqueness result solves a problem left open, even in the non-weighted case, in Vazquez (J Eur Math Soc 16:769–803, 2014).