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

Fabio Punzo,Gabriele Grillo,Matteo Muratori

doi:10.3934/dcds.2015.35.5927

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.

Highlights

We investigate the asymptotic behaviour, as t → ∞, of nonnegative solutions to the following parabolic nonlinear, degenerate, nonlocal weighted problem: ρ(x)ut + (−∆)s(um) = 0 in Rd × (0, ∞), u = u0 on Rd × {0}, (1)where the initial datum u0 is nonnegative and belongs toL1ρ(Rd) = u : u 1,ρ = |u(x)| ρ(x)dx < ∞Rd and the weight ρ is assumed to be positive, locally essentially bounded away from zero (namely ρ−1 ∈ L∞ loc(Rd)) and to satisfy suitable decay conditions at infinity, which we shall specify later.As for the parameters involved, we shall assume throughout the paper that m > 1 and d > 2s
We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density
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

Summary

Introduction

The main goal of this paper is to study the large time behaviour of solutions to problem (1) To this end, to the results recalled above in the local case, we shall distinguish two situations: i) ρ(x) → 0 slowly as |x| → ∞, in the sense that for a suitable γ ∈ (0, 2s) there holds lim ρ(x)|x|γ = c∞ > 0 ;. (ii) Thanks to the results of [23, Section 3.1] (which in turn go back to [14, Section 8.1]), or to the discussion in Appendix A – Part I (which applies to slowly decaying densities as well), we have that the solutions provided by Theorem 2.5 are strong They belong to C((0, ∞); L1ρ(Rd)). (iii) For d ≥ 4s the assumptions of Theorem 2.5 on γ amount to γ ∈ [0, 2s)

Main results

Asymptotic behaviour for rapidly decaying densities

Asymptotic behaviour for slowly decaying densities