Long-time behavior for the porous medium equation with small initial energy

Abstract

We study the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary. 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 point out that the initial datum is allowed to be sign-changing.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.