METASTABLE PATTERNS FOR THE CAHN-HILLIARD EQUATION .2. LAYER DYNAMICS AND SLOW INVARIANT MANIFOLD

In this paper we study the dynamics of the 1-dimensional Cahn-Hilliard equation u(t)=(-epsilon(2)u(xx) + W-n(u))(xx) in a finite interval in a neighborhood of an equilibrium with N+1 transition layers, where epsilon 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(epsilon ln 1/epsilon). 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 Equations 111 (1994), 421-457, with its global Lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in epsilon. (C) 1995 Academic Press, Inc.