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The paper is devoted to a modification of the classical Cahn-Hilliardequation proposed by some physicists. This modification is obtained by addingthe second time derivative of the order parameter multiplied by an inertialcoefficient which is usually small in comparison to the other physicalconstants. The main feature of this equation is the fact that even a globallybounded nonlinearity is "supercritical" in the case of two and three spacedimensions. Thus the standard methods used for studying semilinear hyperbolicequations are not very effective in the present case. Nevertheless, we haverecently proven the global existence and dissipativity of strong solutions inthe 2D case (with a cubic controlled growth nonlinearity) and for the 3D casewith small inertial coefficient and arbitrary growth rate of the nonlinearity.The present contribution studies the long-time behavior of rather weak (energy)solutions of that equation and it is a natural complement of the results of ourprevious papers. Namely, we prove here that the attractors for energy andstrong solutions coincide for both the cases mentioned above. Thus, the energysolutions are asymptotically smooth. In addition, we show that the non-smoothpart of any energy solution decays exponentially in time and deduce that the(smooth) exponential attractor for the strong solutions constructed previouslyis simultaneously the exponential attractor for the energy solutions as well.

Trajectory and smooth attractors for Cahn-Hilliard equations with inertial term