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

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.