Large-time behavior of one-phase Stefan-type problems with anisotropic diffusion in periodic media

TL;DR

This paper investigates the long-term behavior of solutions to a one-phase Stefan problem with anisotropic diffusion in periodic media, demonstrating homogenization of the free boundary velocity and convergence to a self-similar profile.

Contribution

It introduces a rescaling method that shows homogenization of both the free boundary velocity and the anisotropic operator, leading to convergence results for the rescaled solutions.

Findings

Homogenization of free boundary velocity in anisotropic media

Convergence of rescaled solutions to a Hele-Shaw-type problem

Rescaled free boundary approaches a self-similar profile

Abstract

We study the large-time behavior of solutions of a one-phase Stefan-type problem with anisotropic diffusion in periodic media on an exterior domain in a dimension $n \geq 3$. By a rescaling transformation that matches the expansion of the free boundary, we deduce the homogenization of the free boundary velocity together with the homogenization of the anisotropic operator. Moreover, we obtain the convergence of the rescaled solution to the solution of the homogenized Hele-Shaw-type problem with a point source and the convergence of the rescaled free boundary to a self-similar profile with respect to the Hausdorff distance.