We consider the Cahn-Hilliard equation in H 1 (ℝ N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as |u| → ∞ and logistic type nonlinearities. In both situations we prove the H 2 (ℝ N )-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).
Global solvability and asymptotics of semilinear parabolic Cauchy problems in $\mathbb R^n$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over $\mathbb R^n$, $n\in \mathbb N$. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.