Long-time behavior for local and nonlocal porous medium equations with small initial energy

In the first part of the talk, we will describe some aspects of a study developed in a joint paper with L. Brasco concerning the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary and sign-changing initial datum. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant-sign solution of a sublinear Lane-Emden equation, once suitably rescaled.
We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one. The second part of the talk will be devoted to some new advances obtained in collaboration with G. Franzina, in the spirit of the ones explained above, for the study of the asymptotics of signed solutions for the Fractional Porous Medium Equation.