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.
For the equation y (n) + |y| k sgn y = 0, k > 1, n = 3, 4, existence of oscillatory solutions y = (x ∗ − x) −α h(log(x ∗ − x)), α = n ⁄ k − 1 , x < x∗ , is proved, where x ∗ is an arbitrary point and h is a periodic non-constant function on R. The result on existence of such solutions with a positive periodic non-constant function h on R is formulated for the equation y (n) = |y| k sgn y, k > 1, n = 12, 13, 14.
Various new criteria for the oscillation of nonlinear neutral difference equations of the form Δi (xn — x n - h) + qn\xn~g\c sgns n -9 =0 , i = 1,2,3 and c > 0, are established.