  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.
