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

Matteo Bonforte (1) , Yannick Sire (2) , Juan Luis Vazquez (3)

1 Departamento de Matemáticas
2 I2M - Institut de Mathématiques de Marseille
3 Departamento de Matemáticas [Madrid]

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.

Domaines: Equations aux dérivées partielles [math.AP]

Matteo Bonforte, Yannick Sire, Juan Luis Vazquez. Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains. 2014. ⟨hal-01005326⟩