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 Ωᵈ $(d=2,3)$, subject to a kinetic rate dependent nonlocal dynamic boundary condition. The kinetic rate $1/L$, with L∈[0,+∞), distinguishes different types of bulk-surface interactions. When L∈[0,+∞), for a general class of singular potentials including the physically relevant logarithmic potential, we establish the existence of a global attractor AₘL in a suitable complete metric space. Moreover, we verify that the global attractor Aₘ⁰ is stable with respect to perturbations AₘL for small $L>0$. For the case L∈(0,+∞), based on the strict separation property of solutions, we prove the existence of exponential attractors through a short trajectory type technique, which also yields that the global attractor has finite fractal dimension. Finally, when L∈(0,+∞), by usage of a generalized Łojasiewicz-Simon inequality and an Alikakos-Moser type iteration, we show that every global weak solution converges to a single equilibrium in L^∞ as time tends to infinity.