The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere \(\mathbb {S}^2\), at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

本文的主要目的是对一维Landau-Lifshitz-Gilbert方程（LLG）的自收缩解进行分析研究，该模型描述了铁磁材料中自旋的动力学。我们表明，对于LLG方程，存在一个唯一的光滑向后自相似解类，直至对称，并且我们建立了它们的渐近性。此外，我们获得了在存在阻尼的情况下，自相似轮廓的轨迹以指数速率收敛到球面\（\ mathbb {S} ^ 2 \）上的大圆上。特别是，本文提供的结果提供了在有限时间内爆炸的示例，其中由于形成极限圆的快速振荡而产生了奇点。