Long-time behavior of the Cahn–Hilliard equation with dynamic boundary condition

Abstract

We study the long-time behavior, within the framework of infinite dimensional dynamical systems, of the Cahn–Hilliard equation endowed with a new class of dynamic boundary conditions. The system under investigation was recently derived by Liu–Wu (Arch Ration Mech Anal 233:167–247, 2019) via an energetic variational approach such that it naturally fulfills physical properties like mass conservation, energy dissipation and force balance. For the system with regular potentials, we prove the existence of exponential attractors, which also yields the existence of a global attractor with finite fractal dimension. For the system with singular potentials, we obtain the existence of a global attractor in a suitable complete metric space.