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A note on the Cahn-Hilliard equation in $$H^1(\mathbb R^N)$$ involving critical exponent. (English) Zbl 1340.35107
Summary: We consider the Cahn-Hilliard equation in $$H^1(\mathbb R^N)$$ with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as $$| u| \to \infty$$ and logistic type nonlinearities. In both situations we prove the $$H^2(\mathbb R^N)$$-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 and A. Rodriguez-Bernal [J. Differ. Equations 253, No. 12, 3678–3726 (2012; Zbl 1256.35133)].
##### MSC:
 35K30 Initial value problems for higher-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B33 Critical exponents in context of PDEs
Zbl 1256.35133
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