Large time behavior of fractional porous media equation

TL;DR

This paper investigates the long-term behavior of solutions to the fractional porous medium equation in bounded domains, showing convergence to a sign-specific stationary solution under certain energy conditions.

Contribution

It extends the analysis of fractional porous media equations by establishing convergence criteria and identifying the limit profiles based on initial energy levels.

Findings

Solutions converge to a constant sign solution when initial energy is small.

A nonlocal energetic criterion determines whether the limit is positive or negative.

The study applies to equations with homogeneous Dirichlet boundary conditions.

Abstract

Following the methodology of [Brasco and Volzone, Adv. Math. 2022], we study the long-time behavior for the signed Fractional Porous Medium Equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane-Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.