Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains

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.

arXiv commentKeywords: Fractional Laplace operators, Porous Medium diffusion, Existence and uniqueness theory, Asymptotic behaviour, Fractional Sobolev Spaces

References in corpus (2)

• An extension problem related to the fractional Laplacian
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators

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