Convergence in relative error for the Porous Medium equation in a tube

TL;DR

This paper proves that solutions to the Porous Medium equation in a tube converge in relative error to a self-similar profile over time, providing sharp convergence rates and bounds for the free boundary location.

Contribution

It establishes the convergence in relative error to the Friendly Giant for solutions in a tube, with explicit rates and boundary bounds, extending previous traveling wave results.

Findings

Solutions converge in relative error to the Friendly Giant.

Sharp rates of convergence are established.

Uniform bounds for the free boundary location are provided.

Abstract

Given a bounded domain $D \subset \mathbb{R}^N$ and $m > 1$, we study the long-time behaviour of solutions to the Porous Medium equation (PME) posed in a tube \[ \partial_tu = \Delta u^m \quad \text{ in } D \times \mathbb{R}, \quad t > 0, \] with homogeneous Dirichlet boundary conditions on the boundary $\partial D \times \mathbb{R}$ and suitable initial datum at $t=0$. In two previous works, V\'azquez and Gilding & Goncerzewicz proved that a wide class of solutions exhibit a traveling wave behaviour, when computed at a logarithmic time-scale and suitably renormalized. In this paper, we show that, for large times, solutions converge in relative error to the Friendly Giant, i.e., the unique nonnegative solution to the PME posed in the section $D$ of the tube (with homogeneous Dirichlet boundary conditions) having a special self-similar form. In addition, sharp rates of convergence and uniform bounds for the location of the free boundary of solutions are given.