FINITE-DIMENSIONAL DYNAMICS AND INTERFACES INTERSECTING THE BOUNDARY - EQUILIBRIA AND QUASI-INVARIANT MANIFOLD

In the present paper we consider the Allen-Cahn equation in a class of domains consisting of a rectangular part with two attachments on its sides. We establish the existence of stationary solutions with nearly flat interfaces intersecting orthogonally the boundary of the domain a t its rectangular part. We also show that the stability of these equil ibria depends on the geometry of the domain. Finally we obtain some re sults regarding the dynamics of the Allen-Cahn equation, namely we con struct an approximation of the invariant manifold associated with the equilibria (quasi-invariant manifold). Analysis of the vector field ne ar this manifold suggests that the normal velocity of the flat interfa ces is exponentially small in epsilon.