Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains
Fang Li, Kimie Nakashima · Discrete and Continuous Dynamical Systems · 2011
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.