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

Abstract

We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation. 5927 5928 GABRIELE GRILLO, MATTEO MURATORI AND FABIO PUNZO ASYMPTOTIC BEHAVIOUR FOR THE FRACTIONAL PME 5929 and lim t→∞ t α u(t) − u * M (t) ∞ = 0 .Here, u * M is the self-similar Barenblatt solution of mass

The porous medium equation with measure data on negatively curved Riemannian manifolds

“…In particular, existence and uniqueness results for nonnegative solutions have been established. More recently, we should mention that similar results have been generalized to the fractional porous medium equation [32,13,12]. Furthermore, problem (1.1) with the choice M = H N , namely 2) where H N denotes the N -dimensional hyperbolic space, has lately been addressed in a number of papers.…”

“…Also the L 1 -contractivity inequality (5.5) is a classical fact (see [30,Section 3]). For similar issues involving existence, uniqueness and equivalence of different concepts of solution (in the framework of the fractional porous medium equation), we also refer to [12,Appendix A].…”

Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods

“…which by the way could also have been obtained by linearizing (24). In the case m > m * the conservation of relative mass becomes, after linearization,…”

“…|x| γ dt are finite for all t 1 , t 2 > 0, which is enough in order to give sense to (24) at least in a L 1 loc (R + ) sense.…”

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…The fractional porous medium equation has already been studied in R n and the associated mathematical theory has been developed in several directions and under many aspects, see, e.g., [3], [8], [9], [10], [17], [18] and [52]. Note that in the above situations the fractional Laplacian is defined either through its Fourier transform symbol or by the self-adjointness of ∆, so that it is always a particular case of a fractional power of a sectorial operator (see, e.g., [2,Theorem III.4.6.7]).…”