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

2026/07/30 by Taegyu Kim, Soonsik Kwon, Jeongheon Park
Mathematics · #math.AP #msc:35B44 #msc:35Q55 #msc:35Q41

paper · pdf

29 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We study finite-time blow-up for the one-dimensional focusing mass-critical half-wave equation i∂_tu=|D|u-|u|²u. For even initial data with negative energy and mass slightly above the ground-state mass, we prove the log-log upper bound ‖u(t)‖ _H^1/2\lesssim ((log|log(T-t)|)/(T-t))^1/2 as t\uparrow T. This gives, for the half-wave equation, the same log-log law upper bound as in the mass-critical nonlinear Schrödinger equation. The proof follows a similar strategy developed by Merle and Raphaël, 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 Λ-analytic spaces. Then, in the modulation analysis, we use a local-virial spectral property proved in the companion paper \citePark2026arXiv.

Citations

• Finite-time blow-up for the mass-critical half-wave equation with negative energy
• Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass
• Global Well-Posedness and Soliton Resolution for the Half-Wave Maps Equation with Rational Data
• Weak stability of the sum of two solitary waves for Half-wave equation
• Blowup dynamics for inhomogeneous mass critical half-wave equation
• Nondispersive solutions to the mass critical half-wave equation in two dimensions
• Blow-up dynamics for L²-critical fractional Schrödinger equations
• Blow-up criteria for fractional nonlinear Schrödinger equation
• Traveling waves for the quartic focusing Half Wave equation in one space dimension
• A Two-Soliton with Transient Turbulent Regime for the Cubic Half-wave Equation on The Real Line
• Ill-posedness of the cubic nonlinear half-wave equation and other\n fractional NLS on the real line
• Blowup for fractional NLS
• Type II blow up for the energy supercritical NLS
• The instability of Bourgain-Wang solutions for the L² critical NLS
• Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS
• Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang-Mills problems
• Mean Field Dynamics of Boson Stars

Related

• Finite-time blow-up for the mass-critical half-wave equation with negative energy
• Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass
• Blowup dynamics for inhomogeneous mass critical half-wave equation
• Log-log blow up solutions blow up at exactly m points
• Blowup dynamics for mass critical Half-wave equation in 2D