The existence problem for Cahn-Hilliard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point theorem in the level of approximate ploblem. One of the key point is the assumption for maximal monotone graphs with respect to their domains. Thanks to this, we obtain the existence result of the weak solution by using the passage to the limit.

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Condition imposed on the nonlinear terms of a nonlinear diffusion equation withRobin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the Stefan problem, the Hele-Shaw problem, the porous medium equation and the fast diffusion equation, are included in this class. By characterizing this class of equations as an asymptotic limit of the Cahn-Hilliard systems, the growth condition of the nonlinear term can be improved. In this paper, the existence and uniqueness of the solution are proved. From the physical view point, it is natural that, the Cahn-Hilliard system is treated under the homogeneous Neumann boundary condition. Therefore, the Cahn-Hilliard system subject to the Robin boundary condition looks like pointless. However, at some level of approximation, it makes sense to characterize the nonlinear diffusion equations.