  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.

Analysis of PDEs