2026/07/30 by Jeongheon Park · 1 citation
Mathematics · #math.AP #msc:35B44 #msc:35Q55 #msc:35Q41
121 pages. Computer-assisted verification code, input data, and completed outputs are included as ancillary files and are also available at https://github.com/JeongheonPark-CAP/half-wave-cap-verification
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We study 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 finite-time blow-up and obtain the upper bound ‖u(t)‖ H1/2 \lesssim \frac|log(T-t)|1/4√(T-t) \qquadas t\uparrow T. This is the first finite-time blow-up result for negative-energy solutions to the mass-critical half-wave equation in the near-ground-state regime. The proof uses the modulation analysis developed by Merle--Raphaël \citeMerleRaphael2005AnnMath. For the half-wave equation, the key missing ingredient is a coercivity estimate for a nonlocal quadratic form generated by the scaling direction. Unlike the local NLS, the half-wave equation does not admit the ODE methods used to establish the corresponding coercivity estimate. Instead, we prove the coercivity estimate by an analytic reduction followed by a rigorous computer-assisted proof. The spectral analysis part reduces the coercivity problem to a finite collection of spectral inequalities by combining constrained Morse index arguments with the Birman--Schwinger principle. The computer-assisted part certifies these inequalities by interval arithmetic using a validated approximation of the ground state obtained by compactifying the real line. Together with the modulation analysis, this coercivity theorem gives the finite-time blow-up result.