Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn–Hilliard systems

Abstract

A nonlinear diffusion equation with the Robin 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.