# Limiting problems for a nonstandard viscous Cahn-Hilliard system with dynamic boundary conditions

## TL;DR

This paper analyzes a nonlinear phase-field diffusion system with dynamic boundary conditions, focusing on asymptotic limits and long-term behavior, including convergence as viscosity tends to zero and characterization of omega-limit sets.

## Contribution

It introduces a dynamic boundary condition involving the Laplace-Beltrami operator and studies the asymptotic and long-term behavior of the system, extending previous models.

## Key findings

- Solutions converge to the limit problem as viscosity approaches zero
- The long-time behavior and omega-limit sets are characterized for both positive and zero viscosity
- The dynamic boundary condition models additional surface phase transitions

## Abstract

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.