# Long-time behavior of the one-phase Stefan problem in periodic and random media

## TL;DR

This paper investigates the long-term behavior of the one-phase Stefan problem in heterogeneous media, demonstrating homogenization of free boundary velocity and convergence to a spherical shape in the rescaled solutions.

## Contribution

It introduces a rescaling technique aligned with free boundary evolution to prove homogenization and shape convergence in inhomogeneous media.

## Key findings

- Homogenization of free boundary velocity in inhomogeneous media
- Rescaled solutions converge to a self-similar homogeneous solution
- Rescaled free boundary approaches a sphere in Hausdorff distance

## Abstract

We study the long-time behavior of solutions of the one-phase Stefan problem in inhomogeneous media in dimensions $n \geq 2$. Using the technique of rescaling which is consistent with the evolution of the free boundary, we are able to show the homogenization of the free boundary velocity as well as the locally uniform convergence of the rescaled solution to a self-similar solution of the homogeneous Hele-Shaw problem with a point source. Moreover, by viscosity solution methods, we also deduce that the rescaled free boundary uniformly approaches a sphere with respect to Hausdorff distance.