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 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 (powerlike). 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 Vázquez (J Eur Math Soc 16:769-803, 2014). Mathematics Subject Classification

Fundamental solution and long time behavior of the Porous Medium Equation in hyperbolic space

“…The previous authors also showed that in the presence of highly decaying weights, like ρ(x) ≤ C|x| −γ with γ > n, the long-term behavior of the solutions completely departs from the non-weighted case; indeed, solutions tend to stabilize to a constant value all over the space. This and other surprising results have led to a keen interest in the subject with contributions by many authors, like [18,22,20,29,40,26,27,37]. The properties and long-time behavior of the solutions in the presence of weights that decay in a slower way than |x| −n have been investigated in particular by Reyes and the author in a series of papers [41][42][43].…”

On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density

“…We recall now some well posedness results proved in [23]. In fact, thanks to the theory developed therein, we can guarantee existence and uniqueness of weak solutions to (3) (according to Definition 2.4).…”

“…(i) The smoothing effect (20) can be proved as in [23,Proposition 4.6]. In fact, such proof only relies on the validity of the fractional Sobolev inequality…”

“…Existence and uniqueness of solutions to (8) actually follow from the results established in [23] for the more general problem (3). In particular, they are ensured provided γ ∈ [0, 2s)∩[0, d−2s].…”

Smoothing effects for the filtration equation with different powers

“…As regards weighted porous medium equations, in [34] L q0 -L p smoothing effects (with p ∈ (q 0 , ∞)) were established by only assuming a (spectral-gap) Poincaré inequality, which in general prevents L ∞ regularization. As for the fractional porous medium equation on Euclidean space, we quote [22] and [36], where fractional Gagliardo-Nirenberg-type (or Nash-type) inequalities were used. In [14], the same equation was considered on domains with homogeneous Dirichlet boundary conditions, and smoothing effects were proved by means of smart Green-function techniques.…”