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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2509.10304 |
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| _version_ | 1866911365198774272 |
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| author | Lv, Maoyin Wu, Hao |
| author_facet | Lv, Maoyin Wu, Hao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10304 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials Lv, Maoyin Wu, Hao Analysis of PDEs 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. |
| title | Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.10304 |