Finite-dimensional leading order dynamics for the fast diffusion equation near extinction

TL;DR

This paper develops a finite-dimensional approximation framework for the fast diffusion equation near extinction, providing insights into the solution's behavior as it approaches zero.

Contribution

It introduces invariant manifolds that approximate the dynamics near extinction with any desired convergence rate.

Findings

Constructed invariant manifolds for the fast diffusion equation.

Provided finite-dimensional models capturing extinction dynamics.

Achieved prescribed convergence rates near the extinction time.

Abstract

The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near the vanishing solution to any prescribed convergence rate.