JOURNAL OF DIFFERENTIAL EQUATIONS 90, 81-135 (1991) Slow Motion for the Cahn-Hilliard Equation in One Space Dimension NICHOLAS ALIKAKOS* Mathematics Department, The University of Tennessee, Knoxville, Tennessee 37996 PETER W. BATES Mathematics Department, Brigham Young University, Provo, Utah 84602 AND GIORGIO Fusco II University of Rome, Rome, Italy Received January 11, 1990 INTRODUCTION In this paper we establish rigorously the existence of some extremely slowly evolving solutions of the Cahn-Hilliard equation t4, = ( -&zu,, + W(u)),, (CW on 0 < x c 1, subject to the boundary conditions ll,=tt xxx =o at x=0, 1, where W is a smooth double-well function. See Fig. 1. The typical underlying physical context involves a melted binary alloy with a given concentration of components that is quenched to a tem- perature at which exactly two different concentration phases can coexist. The subsequent evolution roughly divides into two stages: a relatively fast one during which the sample becomes inhomogeneous, a fine-grained mixture of the two phases,followed by a very slow coarsening processduring which the originally fine-grained structure becomes less line; always the average concentration remaining constant (for example, [G-D, L]). One * Supported in part by DMS-8804631 and by the Science Alliance, a program at the University of Tennessee Centers of Excellence. 81 OO22-0396/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights 01 reproduction in any fom~ reserved.