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

TL;DR

This paper studies nonlinear diffusion equations with Robin boundary conditions as limits of Cahn-Hilliard systems, improving growth conditions and establishing existence and uniqueness of solutions.

Contribution

It characterizes a class of nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems with Robin boundary conditions, enhancing understanding of boundary effects.

Findings

Established existence and uniqueness of solutions.

Characterized nonlinear diffusion equations as asymptotic limits.

Improved growth conditions for nonlinear terms.

Abstract

Condition imposed on the nonlinear terms of a nonlinear diffusion equation with {R}obin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the {S}tefan problem, the {H}ele--{S}haw 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 {C}ahn--{H}illiard 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 {C}ahn--{H}illiard system is treated under the homogeneous {N}eumann boundary condition. Therefore, the {C}ahn--{H}illiard system subject to the {R}obin boundary condition looks like pointless. However, at some level of approximation, it makes sense to characterize the nonlinear diffusion equations.