Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass
Abstract
We study the blow-up dynamics for the -critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Rapha\"el, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-time blow-up solutions with mass slightly exceeding the threshold. This is inspired by similar results in the mass-critical NLS by Bourgain and Wang, and their instability by Merle, Rapha\"el and Szeftel. We exhibit a blow-up profile driven by the rescaled ground state, with a decoupled dispersive radiation component. We rigorously describe the asymptotic behavior of such solutions near the blow-up time, including sharp modulation dynamics. Furthermore, we demonstrate the instability of these solutions by constructing non-blow-up solutions that are arbitrarily close to the blow-up solutions. The main contribution of this work is to overcome the nonlocal setting of half-wave and to extend insights from the mass-critical NLS to a setting lacking pseudo-conformal symmetry.
Cite
@article{arxiv.2508.07787,
title = {Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass},
author = {Jeongheon Park and Soonsik Kwon and Taegyu Kim},
journal= {arXiv preprint arXiv:2508.07787},
year = {2025}
}