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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)].
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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