A General Fractional Porous Medium Equation

Abstract

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,We consider data f ∈ L 1 (R N ) and all exponents 0 < σ < 2 and m > 0. Existence and uniqueness of a weak solution is established for m > m * = (N − σ) + /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 ≤ 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 σ. Moreover, we consider the above questions for the problem posed in a bounded domain.2000 Mathematics Subject Classification. 26A33, 35A05, 35K55, 76S05

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

“…One proceeds picking a sequence of nonnegative data u 0n ∈ L 1 ρ (R d ) ∩ L ∞ (R d ) such that u 0n → u 0 in L 1 ρ (R d ) and pass to the limit in (11) as n → ∞ by exploiting (113), (120) and (123) for p = 1 (see also [32, Theorem 6.5 and Remark 6.11]). Such solutions are still strong because the L 1 ρ comparison principle (113) is preserved (which is in fact one of the main tools to prove that solutions are strong -see again [14,Section 8.1] and references quoted).…”

“…Furthermore, by suitably exploiting the celebrated Stroock-Varopoulos inequality (see [14,Proposition 8.5] or [23, Section 4.2]), one can show that for any p ∈ [1, ∞] the L p ρ norm of u(t) does not increase in time. Now suppose that, in addition to the above hypotheses, ρ ∈ L ∞ (R d ).…”

“…[13,14,21,32]), are discussed in Appendix A. The first result we provide reads as follows (for a sketch of proof see again Appendix A -Parts I and II).…”

Sharp short and long timeL∞bounds for solutions to porous media equations with homogeneous Neumann boundary conditions

“…In fact, for notational simplicity, we shall give the complete proof only for q 0 = 1. We shall exploit a technique similar to the one used in the proof of Corollary 8.1 from [25]. That is, first of all consider the analogue of estimate…”

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.…”