On a class of Cahn-Hilliard models with nonlinear diffusion. (English) Zbl 1276.35101

A class of Cahn-Hilliard equations characterized by a nonlinear diffusive dynamics and possibly containing an additional sixth order term is considered. This model describes the separation properties of oil-water mixtures when a substance enforcing the mixing of the phases (a surfactant) is added. However, the model is also closely connected with other Cahn-Hilliard-like equations relevant in different types of applications. The authors discuss the existence of a weak solution to the sixth order model in the case when the configuration potential of the system has a singular (e.g., logarithmic) character. The behavior of the solutions in the case when the sixth order term is allowed to tend to \(0,\) proving convergence to solutions of the fourth order system in a special case. The fourth order system is investigated by a direct approach, and existence of a weak solution is shown under very general conditions by means of a fixed point argument. Additional properties of the solutions, like uniqueness and parabolic regularization, are discussed, both for the sixth order and for the fourth order model, under more restrictive assumptions on the nonlinear diffusion term.


35K35 Initial-boundary value problems for higher-order parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
47H05 Monotone operators and generalizations
35K58 Semilinear parabolic equations
35D30 Weak solutions to PDEs
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