数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 1751-1768.
有界域上四阶 Cahn-Hilliard 方程在二重特征值处的定态分歧
廖睿琳(
), 管芳(), 潘志刚*()
- 西南交通大学数学学院 成都 611756
-
收稿日期:2025-04-27修回日期:2025-10-21出版日期:2026-10-26发布日期:2026-09-07 -
通讯作者:潘志刚 E-mail:liaorl02@126.com;15528026689@163.com;panzhigang@home.swjtu.edu.cn -
作者简介:廖睿琳, E-mail:liaorl02@126.com;
管芳, E-mail:15528026689@163.com -
基金资助:国家自然科学基金(11901408);国家自然科学基金(7251223);四川省自然科学青年基金(22NSFSC16338);中央高校理科创新培育项目(2682026CX185)
Steady-State Bifurcation From Double Eigenvalue for the Fourth-Order Cahn-Hilliard Equation on a Bounded Domain
Ruilin Liao(), Fang Guan(), Zhigang Pan*()
- School of Mathematics, Southwest Jiaotong University, Chengdu 611756
-
Received:2025-04-27Revised:2025-10-21Online:2026-10-26Published:2026-09-07 -
Contact:Zhigang Pan E-mail:liaorl02@126.com;15528026689@163.com;panzhigang@home.swjtu.edu.cn -
Supported by:NSFC(11901408);NSFC(7251223);Sichuan Provincial Natural Science Youth Fund(22NSFSC16338);Central University Basic Research Innovation Project(2682026CX185)
摘要：
该文运用规范化的 Lyapunov-Schmidt 约化方法和线性全连续场谱分解定理, 研究 Cahn-Hilliard 方程的定态分歧. 在正方形有界域上, 作者证明了四阶 Cahn-Hilliard 方程在齐次 Dirichlet 边界条件下和齐次 Robin 边界条件下在第一特征值 (二重) 处发生定态分歧, 此时 Cahn-Hilliard 方程有非平凡解, 并得到了两类边界条件下产生超临界分歧和次临界分歧的完整判据、分歧解的具体表达式、正则性以及分歧解图.
中图分类号:
- O175.29
引用本文
廖睿琳, 管芳, 潘志刚. 有界域上四阶 Cahn-Hilliard 方程在二重特征值处的定态分歧[J]. 数学物理学报, 2026, 46(5): 1751-1768.
Ruilin Liao, Fang Guan, Zhigang Pan. Steady-State Bifurcation From Double Eigenvalue for the Fourth-Order Cahn-Hilliard Equation on a Bounded Domain[J]. Acta mathematica scientia,Series A, 2026, 46(5): 1751-1768.
图1
当 $ \alpha_1=1, L=1, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的次临界分歧解"
图2
当 $ \alpha_1=1, L=1, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的超临界分歧解"
图3
当 $ \alpha_1=1, L=2, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的次临界分歧解"
图4
当 $ \alpha_1=1, L=2, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的超临界分歧解"
图5
当 $ \alpha_1=1, L=2, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_2} $ 的次临界分歧解"
图6
当 $ \alpha_1=1, L=2, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_2} $ 的超临界分歧解"
图7
当 $ \alpha_1=1, L=2, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_3} $ 的次临界分歧解"
图8
当 $ \alpha_1=1, L=2, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_3} $ 的超临界分歧解"
图9
当 $ \alpha_1=1, L=1, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的次临界分歧解"
图10
当 $ \alpha_1=1, L=1, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的超临界分歧解"
图11
当 $ \alpha_1=1, L=3, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的次临界分歧解"
图12
当 $ \alpha_1=1, L=3, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_1} $ 的超临界分歧解"
图13
当 $ \alpha_1=1, L=3, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_2} $ 的次临界分歧解"
图14
当 $ \alpha_1=1, L=3, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_2} $ 的超临界分歧解"
图15
当 $ \alpha_1=1, L=3, \lambda<\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_3} $ 的次临界分歧解"
图16
当 $ \alpha_1=1, L=3, \lambda>\frac{\pi^{2}}{L^{2}} $ 时 $ \overline{u_3} $ 的超临界分歧解"
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