On the Cauchy problem of a sixth-order Cahn-Hilliard equation arising in oil-water-surfactant mixtures

Abstract

We study the global well-posedness and asymptotic behavior of solutions for the Cauchy problem of three-dimensional sixth order Cahn-Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the $H^2$-norm of initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the time decay rate of strong solutions. It is worth pointing out that although the problem we considered is a sixth-order parabolic equation, the time decay rate is equivalent to the decay rate of fourth-order generalized heat equation, which is better than our expect.