Metastable Patterns for the Cahn-Hilliard Equation: Part II. Layer Dynamics and Slow Invariant Manifold
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Abstract In this paper we study the dynamics of the 1-dimensional Cahn-Hilliard equation u t =(−ϵ 2 u xx + 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 Equations 111 (1994), 421-457, with its global Lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in ϵ.