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This paper gives theoretical results on spinodal decomposition for the stochastic Cahn-Hilliard-Cook equation, which is a Cahn-Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size.In more detail, our results can be summarized as follows. The Cahn-Hilliard-Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε → 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finitedimensional subspace Y ε is high as long as the solution stays within distance r ε = O(ε R ) of the homogeneous state. The subspace Y ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise.

While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computerassisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczyński [22], and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.

Abstract. We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain G. This equation depends on a small interaction parameter ε > 0, and one is generally interested in asymptotic results as ε → 0. Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach ε −2+dim G/2 with respect to the H 2 (G)-norm.