Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (3): 461-474.

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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
  • Received:2017-10-19 Online:2019-06-26 Published:2019-06-27
  • Contact: Penghong Zhong
  • 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)


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.

Key words: Landau-Lifshitz-Gilbert equation, Global existence, Small solution, Blowup solution

CLC Number: 

  • O175.29