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

Norbert Požár; Giang Thi Thu Vu

doi:10.3934/dcdss.2018058

Article Contents

2018, Volume 11, Issue 5: 991-1010. Doi: 10.3934/dcdss.2018058

This issue Previous Article One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach Next Article

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

Norbert Požár^1, , and
Giang Thi Thu Vu^2,

1.
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan
2.
Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

* Corresponding author: Norbert Požár
* Corresponding author: Norbert Požár

Received: February 2017

Revised: May 2017

Published: June 2018

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Abstract

We study the long-time behavior of solutions of the one-phase Stefan problem in inhomogeneous media in dimensions n ≥ 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.

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Mathematics Subject Classification: 35B27 (35R35, 74A50, 80A22).

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References

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