Asymptotic behaviour of a porous  medium equation with  fractional diffusion

Luis Caffarelli; Juan-Luis Vázquez

doi:10.3934/dcds.2011.29.1393

Article Contents

2011, Volume 29, Issue 4: 1393-1404. Doi: 10.3934/dcds.2011.29.1393

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Asymptotic behaviour of a porous medium equation with fractional diffusion

Luis Caffarelli^1, and
Juan-Luis Vázquez^2,

1.
Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid

Received: March 2010

Revised: August 2010

Published: October 2011

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Abstract

We consider a porous medium equation with a nonlocal diffusion effect given by an inverse fractional Laplacian operator. The equation is posed in the whole space $\mathbb{R}^n$. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy estimates. Here we establish the large-time behaviour. We first find selfsimilar nonnegative solutions by solving an elliptic obstacle problem for the pair pressure-density involving the Laplacian, obtaining what 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, which represent a surprising variation of the Barenblatt profiles of the standard porous medium model.

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Mathematics Subject Classification: Primary: 35B40, 35K55, 35K65, 35J87, 26A33.

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