The dynamics of patterns for two phase separation equations

Two parabolic partial differential equations, the Allen-Cahn equation and the Cahn-Hilliard equation, are studied. Solutions of these equations form spatial patterns that evolve in time.; For the Allen-Cahn equation, the analysis is on an initial value problem that is appropriate for development of the spatial patterns from constant stationary solutions. A model is developed to predict the patterns that form.; For the Cahn-Hilliard equation, the analysis is concentrated on the pattern dynamics once a well formed pattern exists. Discrete and continuous models of the pattern dynamics are considered. Predictions of these models include a period doubling phenomena and a self-similar pattern.; All the results given here are for one spatial dimension. These equations have applications in the material sciences and these applications are discussed.