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

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


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