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.

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