On the Cahn–Hilliard equation in H1(RN)

Abstract

MSC: 35Q55 35K30 35K58 35B40 35B41 Keywords: Cahn-Hilliard equation Initial value problems for higher-order parabolic equations Semilinear parabolic equations Asymptotic behavior of solutions AttractorsIn this paper we exhibit the dissipative mechanism of the Cahn-We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in H 1 (R N ), cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor.