### A note on the Cahn-Hilliard equation in ${H}^{1}\left({\mathbb{R}}^{N}\right)$ involving critical exponent

We consider the Cahn-Hilliard equation in ${H}^{1}\left({\mathbb{R}}^{N}\right)$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $\left|u\right|\to \infty $ and logistic type nonlinearities. In both situations we prove the ${H}^{2}\left({\mathbb{R}}^{N}\right)$-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J. W. Cholewa, A. Rodriguez-Bernal (2012).