Dynamics of the viscous Cahn–Hilliard equation

We study generalized viscous Cahn–Hilliard problems with nonlinearities satisfying critical growth conditions in W 1,p 0 (Ω), where Ω is a bounded smooth domain in Rn, n 3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p = 2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in H1 0 (Ω) and, uniformly with respect to the viscosity parameter, L ∞ (Ω) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n = 3, 4, the attractor of the Cahn–Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation. © 2008 Elsevier Inc. All rights reserved.