Metastable Patterns for the Cahn-Hilliard Equation: Part II. Layer Dynamics and Slow Invariant Manifold

In this paper we study the dynamics of the 1-dimensional Cahn-Hilliard equation ut=(−ε2uxx+W′(u))xx in a finite interval in a neighborhood of an equilibrium with N+1 transition layers, where ε is a small parameter and W is a double well energy density function with equal minima. The lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the Cahn-Hilliard equation starts outside a neighborhood of the equilibrium of size O(ε ln 1/ε). It is proved that there is an N-dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in J. Differential Equations111 (1994), 421-457, with its global Lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in ε.