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.

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

Cited by in corpus (10)

• Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods
• Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
• Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions
• Compactness of Green operators with applications to semilinear nonlocal elliptic equations
• The evolution fractional p-Laplacian equation in ${\mathbb{R}}^N$. Fundamental solution and asymptotic behaviour
• An inverse problem for the fractional porous medium equation
• Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded Euclidean domain
• Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradient
• Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations
• A doubly nonlinear evolution problem involving the fractional p-Laplacian