Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials

Abstract

We investigate the long-time behavior of a nonlocal Cahn-Hilliard equation in a bounded domain $\Omega \subset {\mathbb{R}}^d$ $(d\in \{2,3\})$, subject to a kinetic rate-dependent nonlocal dynamic boundary condition. The kinetic rate $1/L$, with $L\in [0,+\infty )$, distinguishes different types of bulk-surface interactions. For general singular potentials, including the physically relevant logarithmic potential, we establish the existence of a global attractor ${\mathcal{A}}_m^L$ in a suitable complete metric space for any $L\in [0,+\infty )$. Moreover, we verify that the global attractor ${\mathcal{A}}_m^0$ is stable with respect to perturbations ${\mathcal{A}}_m^L$ for small $L>0$. When $L\in (0,+\infty )$, based on the strict separation property of global weak solutions, we further prove the existence of exponential attractors via a short-trajectory type technique, which also implies that the global attractor has finite fractal dimension. Finally, for this case, we show that every global weak solution converges to a single equilibrium in ${\mathcal{L}}^{\infty }$ as time goes to infinity, using a generalized Łojasiewicz-Simon inequality and an Alikakos-Moser type iteration.