Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains

Fang Li; Kimie Nakashima

doi:10.3934/dcds.2012.32.1391

Article Contents

2012, Volume 32, Issue 4: 1391-1420. Doi: 10.3934/dcds.2012.32.1391

This issue Previous Article Solutions to a fluid-structure interaction free boundary problem Next Article Dominated splitting and Pesin's entropy formula

Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains

Fang Li^1, and
Kimie Nakashima^2,

1.
Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai, 200241
2.
Tokyo University of Marine Science and Technology, 4-5-7 Konan, Minato-ku, Tokyo 108-8477

Received: September 30, 2010

Revised: July 31, 2011

Published: March 2012

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Abstract

In this paper, we study a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. By upper and lower solution method, we obtain a sufficient condition for a hypersurface $S$ in the domain $\Omega$ to support stable transition layers, and a necessary condition for $S$ in $\Omega$ to support transition layers, not necessarily stable. In addition, sharp estimates on depths of transition layers have also been derived.

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Mathematics Subject Classification: Primary: 35J25, 35B25; Secondary: 35B35.

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