A log-log upper bound on blow-up rates for the mass-critical half-wave equation

Abstract

We study finite-time blow-up for the one-dimensional focusing mass-critical half-wave equation \begin{equation*} i\partial_tu=|D|u-|u|^2u. \end{equation*} For even initial data with negative energy and mass slightly above the ground-state mass, we prove the log-log upper bound \begin{equation*} \|u(t)\|_{\dot H^{1/2}}\lesssim \left(\frac{\log|\log(T-t)|}{T-t}\right)^{1/2} \quad \text{as}\quad t\uparrow T. \end{equation*} This gives, for the half-wave equation, the same log-log law upper bound as in the mass-critical nonlinear Schr\"odinger equation. The proof follows a similar strategy developed by Merle and Rapha\"el, but requires a new construction of the blow-up profile. Main difficulty arises from the nonlocal operator $|D|$ and the absence of pseudo-conformal symmetry. We construct an almost self-similar profile with exponentially small error by combining tail computations carried out to arbitrary order, depending on a dynamical parameter, with Borel integral summation in $\Lambda$-analytic spaces. Then, in the modulation analysis, we use a local-virial spectral property proved in the companion paper \cite{Park2026arXiv}.