Attractors for some models of the Cahn-Hilliard equation with the inertial term

In this work, we consider the initial boundary value problem for the modified version of the Cahn-Hilliard equation with the inertial term in two dimensional smooth bounded domain. Under the optimal regularity condition on the quartic nonlinearity, by using the splitting method, we prove the existence of the absorbing set for the weak solutions. Then, applying the energy method, we show that the semigroup generated by the weak solutions possesses a global attractor of the optimal regularity, and thereby we give a positive answer to the question raised in (Grasselli et al 2009 On the 2D Cahn-Hilliard equation with inertial term Commun. PDE 34 137-70).