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We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape & Miller, 2002] with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path-following of the parameter range of Cahn–Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we, for example, find nonunique connection matrices and saddle–saddle connections of Morse sets. [ABSTRACT FROM AUTHOR]

This paper discusses the roughness of surfaces described by nonlinear stochastic partial differential equations on bounded domains. Roughness is an important characteristic for processes arising in molecular beam epitaxy, and is usually described by the mean interface width of the surface, i.e. the expected value of the squared Lebesgue norm. By employing results on the mean interface width for linear stochastic partial differential equations perturbed by colored noise, which have been previously obtained, we describe the evolution of the surface roughness for two classes of nonlinear equations, asymptotically both for small and large times. [ABSTRACT FROM AUTHOR]