AI 中文总结
该研究针对带负能量、质量略高于基态质量的偶初始数据，结合调制分析与解析约化及计算机辅助证明，得到一维质量临界半波方程的首个近基态区域负能量解的有限时间爆破结果。
AI 中文摘要
我们研究一维聚焦型质量临界半波方程 \\( i\partial_tu=|D|u-|u|^2u \\)。对于具有负能量且质量略高于基态质量的偶初始数据，我们证明了有限时间爆破，并得到当 \\( t\uparrow T \\) 时的上界 \\( \\|u(t)\\|_{\dot H^{1/2}} \lesssim \frac{|\log(T-t)|^{1/4}}{\sqrt{T-t}} \\)。这是近基态区域内质量临界半波方程负能量解的首个有限时间爆破结果。证明使用了Merle–Raphaël提出的调制分析方法，对于半波方程，关键缺失部分是尺度方向产生的非局部二次型的 coercivity（强制性）估计。与局部非线性薛定谔方程（NLS）不同，半波方程不具备建立对应强制性估计所用的常微分方程（ODE）方法，因此我们通过解析约化结合严格的计算机辅助证明来推导该强制性估计。谱分析部分通过结合约束Morse指标论证与Birman–Schwinger原理，将强制性问题约化为有限组谱不等式；计算机辅助部分通过区间算术，利用紧致化实直线得到的基态的验证近似来证明这些不等式。结合调制分析，该强制性定理给出了有限时间爆破结果。
英文摘要
We study the one-dimensional focusing mass-critical half-wave equation \[ i\partial_tu=|D|u-|u|^2u. \] 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 \[ \|u(t)\|_{\dot H^{1/2}} \lesssim \frac{|\log(T-t)|^{1/4}}{\sqrt{T-t}} \qquad\text{as }t\uparrow T. \] 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 \cite{MerleRaphael2005AnnMath}. 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.
Comments121 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
