Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains

Matteo  Bonforte; Yannick  Sire; Juan Luis  Vázquez

doi:10.3934/dcds.2015.35.5725

Article Contents

2015, Volume 35, Issue 12: 5725-5767. Doi: 10.3934/dcds.2015.35.5725

This issue Previous Article Eventual regularity for the parabolic minimal surface equation Next Article On the variation of the fractional mean curvature under the effect of $C^{1, \alpha}$ perturbations

Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid
2.
Université Aix-Marseille, I2M, Centre de Mathématique et Informatique, Technopôle de Chateau-Giombert, Marseille

Received: March 31, 2014

Published: November 2015

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Abstract

We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

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Mathematics Subject Classification: Primary: 35K55, 35K61, 35K65, 35B40, 35A01, 35A02.

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