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

TL;DR

This paper investigates the long-term behavior of solutions to the porous medium equation with small initial energy, demonstrating convergence to a sign-specific steady state under certain conditions.

Contribution

It establishes conditions under which solutions with sign-changing initial data converge to a nontrivial steady state, extending understanding of the porous medium equation's asymptotic behavior.

Findings

Solutions with small initial energy converge to a steady state.

Convergence depends on the initial energy and sign of the data.

A criterion determines whether the limit is positive or negative.

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.