• 论文 •

### 双曲空间上的Landau-Lifshitz-Gilbert方程解的全局存在性与自相似爆破解

1. 1 广东第二师范学院数学系 广州 510640
2 云南民族大学数学系 昆明 650031
3 云南师范大学数学科学学院 昆明 650092
• 收稿日期:2017-10-19 出版日期:2019-06-26 发布日期:2019-06-27
• 通讯作者: 钟澎洪 E-mail:gzydshang@126.com
• 基金资助:
国家自然科学基金青年基金(11601092);广东省青年创新人才项目基金(2014KQNCX228);广东省科技厅博士启动基金(2014A030310330);广州市科技计划项目基金(201607010352)

### Global Existence and Self-Similar Blowup of Landau-Lifshitz-Gilbert Equation on Hyperbolic Space

Penghong Zhong1,*(),Ganshan Yang2,Xuan Ma3

1. 1 Department of Mathematics, Guangdong University of Education, Guangzhou 510640
2 Department of mathematics, Yunnan Nationalities University, Kunming 650031
3 Institute of Mathematics Science, Yunnan Normal University, Kunming 650092
• Contact: Penghong Zhong E-mail:gzydshang@126.com
• Supported by:
the National Science Foundation for Young Scientists of China(11601092);the Project for Young Creative Talents of Ordinary University of Guangdong Province(2014KQNCX228);the PhD Start-up Fund of Natural Science Foundation of Guangdong Province(2014A030310330);the Funds of Guangzhou Science and Technology(201607010352)

Abstract:

By the generalized Hasimoto transformation, we deduce an equivalent system of the Landau-Lifshitz-Gilbert equation on hyperbolic space $\mathbb{H}$2. Based on this equivalent model, we prove the global existence of the Landau-Lifshitz-Gilbert equation with the small initial condition. Until now, we have not seen a paper discussing the explicit dynamic solution of the complete equation with a damping term on this target. We construct an explicit small data global solution by the equivalent system obtained in this paper. An self-similar finite blowup solution is also presented for the equivalent system. In the previous paper[25], we constructed a finite time blowup solution without Gilbert damping on $\mathbb{H}$2. The question of whether a solution of the complete equation with a Gilbert term can develop a finite time blowup from $\mathbb{H}$2 and smooth initial data is not clear. The self-similar finite time blowup solution we presented here is a finite energy solution on the entire spacial domain. Our result gives a positive answer to this question.

• O175.29