Dynamics of the viscous Cahn–Hilliard equation

AbstractWe study generalized viscous Cahn–Hilliard problems with nonlinearities satisfying critical growth conditions in W01,p(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 H01(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