AI 中文总结
针对一维聚焦质量临界半波方程，研究具有负能量且质量略高于基态质量的偶初始数据的有限时间爆破，构造近似自相似剖面并证明其爆破率的对数-对数上界，与质量临界非线性薛定谔方程的结果一致。
AI 中文摘要
我们研究一维聚焦质量临界半波方程的有限时间爆破，其方程为$i\partial_tu=|D|u-|u|^2u$。对于具有负能量且质量略高于基态质量的偶初始数据，我们证明了对数-对数上界：当$t$趋近于$T$时，$\\|u(t)\\|_{\dot H^{1/2}}\lesssim \left(\frac{\log|\log(T-t)|}{T-t}\right)^{1/2}$。这为半波方程给出了与质量临界非线性薛定谔方程相同的对数-对数律上界。证明遵循Merle和Raphaël提出的类似策略，但需要新的爆破剖面构造。主要困难源于非局部算子$|D|$和伪共形对称性的缺失。我们结合对任意阶进行的尾项计算（依赖于动力学参数）与$\Lambda$-解析空间中的Borel积分求和，构造了具有指数小误差的近似自相似剖面。随后在调制分析中，我们使用了在配套论文\cite{Park2026arXiv}中证明的局部virial谱性质。
英文摘要
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 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 \cite{Park2026arXiv}.
Comments29 pages
