Calculus of Variations and Geometric Measure Theory

L. Brasco - B. Volzone

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

created by brasco on 07 Nov 2020
modified on 12 Aug 2021


Accepted Paper

Inserted: 7 nov 2020
Last Updated: 12 aug 2021

Journal: Adv. Math.
Pages: 41
Year: 2021


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.

Keywords: asymptotic behavior, Lane-Emden equation, Porous medium equation