Asymptotic Analysis of the Allen–Cahn Equation with Dynamic Boundary Conditions of Cahn–Hilliard Type

Problems for partial differential equations coupled with dynamic boundary conditions can be viewed as a type of transmission problem between the bulk and its boundary. For the heat equation and the Allen–Cahn equation, various forms of such problems with dynamic boundary conditions are studied in this paper. In the case of the Cahn–Hilliard equation in the bulk, several models have been proposed in which the boundary equations and conditions differ. Recently, the vanishing surface diffusion limit has been investigated in more than one of these models. In such settings, the resulting dynamic boundary equation typically takes the form of a forward–backward parabolic equation. In this paper, we focus on a different model, in which the Allen–Cahn equation governs the bulk dynamics, while the boundary condition is of Cahn–Hilliard type. We analyze the asymptotic behavior of the system, including the well-posedness of the limiting problems and corresponding error estimates for the differences between solutions. These aspects are discussed for three types of limiting systems.