Finite-time singularities in the one-dimensional half-harmonic map heat flow

Determine whether finite-time singularities can occur for the half-harmonic map heat flow from \(\mathbb{R}\) into \(\mathbb{S}^1\).

Background

The paper discusses the long-time behavior of the half-harmonic map heat flow in the one-dimensional setting. Although infinite-time blow-up solutions have been constructed for the flow from $\mathbb{R}$ into $\mathbb{S}^1$, the authors explicitly identify the occurrence of finite-time singularities as unresolved. This question concerns whether solutions can develop singularities after a finite duration of evolution, complementing the existing analysis of bubbling and singular times in related settings.