METASTABLE PATTERNS FOR THE CAHN-HILLIARD EQUATION .2. LAYER DYNAMICSAND 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 ep silon is a small parameter and W is a double well energy density funct ion with equal minima. The lower bound of the layer motion speed is gi ven explictly and the layer motion directions are described precisely if a solution of the Cahn-Hilliard equation starts outside a neighborh ood of the equilibrium of size O(epsilon ln 1/epsilon). It is proved t hat there is an N-dimensional unstable invariant manifold which is a s mooth graph over the approximate manifold constructed in J. Differenti al Equations 111 (1994), 421-457, with its global Lipschitz constant e xponentially small and this unstable invariant manifold attracts solut ions exponentially fast uniformly in epsilon. (C) 1995 Academic Press, Inc.