Invariant manifolds for the porous medium equation

TL;DR

This paper constructs invariant manifolds for the porous medium equation to analyze the long-term behavior and higher-order asymptotics of solutions, using a nonlinear change of variables and dynamical systems techniques.

Contribution

It introduces a novel framework involving invariant manifolds on a hypocycloidic manifold to study the equation's asymptotics and solution dependence on initial data.

Findings

Invariant manifolds are constructed tangent to eigenspaces at the origin.

The framework allows for detailed higher-order asymptotic expansions.

Solutions depend differentiably on initial data within this framework.

Abstract

In this paper, we investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation posed in $\mathbf{R}^N$. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a fixed domain with quadratic nonlinearity. The degeneracy is cured by viewing the dynamics on a hypocycloidic manifold. It is in this framework that we can prove a differentiable dependency of solutions on the initial data, and thus, dynamical systems methods are applicable. Our main result is the construction of invariant manifolds in the phase space of solutions which are tangent at the origin to the eigenspaces of the linearized equation. We show how these invariant manifolds can be used to extract information on the higher-order long-time asymptotic expansions of solutions to the porous medium equation.