Metastable patterns for the Cahn-Hilliard equation

The dynamics of the one-dimensional Cahn-Hilliard equation in a neighborhood of the unstable manifold of an equilibrium having N + 1 transition layers are studied. An approximation for an N dimensional invariant manifold and "slow channel" are defined and it is proved that if a solution of the Cahn-Hilliard equation starts outside but close to the slow channel of the approximate manifold, then it will approach the channel at an exponentially large speed. After it enters the slow channel, it will follow the approximate manifold with speed {dollar}O(esp{lcub}-c/epsilon{rcub}){dollar}, where {dollar}c > 0{dollar} and {dollar}0 < epsilon <<1,{dollar} and stay in the channel for exponentially large time. Bounds on the layer motion speed are given explicitly and proved to be exponentially small in terms of {dollar}epsilon{dollar} 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 {dollar}O(epsilonell n {lcub}1overepsilon{rcub}){dollar}. It is proved that there is an N-dimensional unstable manifold which is a smooth graph over the approximate manifold with its global Lipschitz constant exponentially small and this unstable manifold attracts solutions at a speed {dollar}O(esp{lcub}-ct{rcub}){dollar} for some constant c independent of {dollar}epsilon{dollar} before the solution leaves the channel through the ends.