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

TL;DR

This paper studies fractional porous medium equations on bounded domains, proving existence, uniqueness, and long-term behavior of solutions, and extends results to related elliptic equations with sub-linear nonlinearities.

Contribution

It establishes the first comprehensive existence, uniqueness, and asymptotic analysis for fractional porous medium equations on bounded domains using maximal monotone operator theory.

Findings

Existence and uniqueness of solutions for fractional porous medium equations.

Long-time asymptotic behavior characterized by friendly giant solutions.

Extension of results to semilinear elliptic nonlocal equations.

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.