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

Gabriele Grillo; Matteo  Muratori; Fabio  Punzo

doi:10.3934/dcds.2015.35.5927

Article Contents

2015, Volume 35, Issue 12: 5927-5962. Doi: 10.3934/dcds.2015.35.5927

This issue Previous Article Harnack type inequalities for some doubly nonlinear singular parabolic equations Next Article Ground states for scalar field equations with anisotropic nonlocal nonlinearities

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

1.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano
2.
Dipartimento di Matematica , Università di Milano, via Cesare Saldini 50, 20133 Milano

Received: February 28, 2014

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 35B40, 35K55; Secondary: 35C06, 35K65.

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