Large time behavior of fractional porous media equation

Abstract

Following the methodology of [12], 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. Contents 1. Introduction 2. Notations and Assumptions 3. Elliptic Toolkit 3.1. The ground state level 3.2. Higher energies 3.3. Mountain pass level 3.4. First excited level and spectral gap. 4. The (rescaled) parabolic problem 4.1. Well-posedness and entropy-entropy dissipation inequality 4.2. Stabilization 5. Paths of controlled energy 6. Proofs of the main results 6.1. Proof of Theorem 1.1 6.2. Proof of Proposition 1.1 Appendix A. An elementary inequality References