Asymptotic behaviour of a porous medium equation with fractional diffusion

TL;DR

This paper investigates the long-term behavior of solutions to a porous medium equation with nonlocal fractional diffusion, establishing selfsimilar solutions and demonstrating that general solutions tend toward these asymptotic states.

Contribution

It introduces obstacle Barenblatt solutions for the fractional porous medium equation and uses entropy methods to analyze asymptotic behavior, extending previous work on existence.

Findings

Existence of selfsimilar obstacle solutions for the fractional porous medium equation

Asymptotic convergence of general solutions to obstacle Barenblatt solutions

Application of entropy methods to describe large-time behavior

Abstract

We consider a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. The equation is posed in the whole space R^n. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density, which we call obstacle Barenblatt solutions. The theory for elliptic fractional problems with obstacles has been recently established. We then use entropy methods to show that the asymptotic behavior of general finite-mass solutions is described after renormalization by these special solutions.