Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials
Maoyin Lv, Hao Wu
TL;DR
This work analyzes a nonlocal Cahn–Hilliard system in a bounded domain with a nonlocal dynamic boundary condition, incorporating a relaxation parameter $L$ and singular potentials. It develops a unified functional-analytic framework to treat bulk–surface coupling, proves the existence of a global attractor $\mathcal{A}_m^L$ for all $L\ge0$, and establishes upper semicontinuity of the attractors as $L\to0$. For $L>0$, it constructs exponential attractors with finite fractal dimension via a short-trajectory approach, and, when the potentials are real analytic, proves convergence of all global weak solutions to a single equilibrium using a generalized Łojasiewicz–Simon inequality together with smoothing arguments. These results provide a rigorous characterization of the long-time dynamics and attractor structure in bulk–surface phase separation models with singular energies.
Abstract
We investigate the long-time behavior of a nonlocal Cahn-Hilliard equation in a bounded domain $Ω\subset\mathbb{R}^d$ $(d\in\{2,3\})$, 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.