AI 中文总结
本文针对无质量共振的四维质量临界二次非线性薛定谔方程组，采用局部virial论证证明负能径向H¹×H¹解正反方向均有限时间爆破，无需有限方差假设。
AI 中文摘要
我们研究聚焦二次非线性薛定谔方程组：$\begin{cases} i\partial_t u+\Delta u=-2v\overline{u}, \\\\ i\partial_t v+\kappa\Delta v=-u^2, \end{cases}$，其中$\kappa>0$，$(t,x)\in I\times\mathbb R^4$。在非质量共振情形（$\kappa\neq \frac12$），Inui-Kishimoto-Nishimura与Dinh-Forcella的前期工作表明，负能径向解要么在有限时间内爆破，要么整体存在但$H^1$范数无界增长。本文证明：每一个负能径向$H^1\times H^1$解在时间正、反方向均会在有限时间内爆破，且无需有限方差假设。核心方法是基于有界指数权重$\nabla\phi_R(x)=2x e^{-|x|^2/R^2}$的局部virial论证；径向加权插值估计可将非线性误差项控制在对应加权动能项内，误差仅为依赖守恒质量的$O(R^{-2})$；局部virial量本身可直接由同一加权动能缺陷界定，结合这些估计得到超线性Riccati型微分不等式，该不等式无法持续存在，故迫使有限时间爆破。
英文摘要
We study the focusing quadratic nonlinear Schrödinger system \[ \begin{cases} i\partial_t u+Δu=-2v\overline{u}, \\ i\partial_t v+κΔv=-u^2, \end{cases} \qquad (t,x)\in I\times\mathbb R^4, \] where $κ>0$. In the non-mass-resonant case $κ\neq \frac12$, previous works of Inui--Kishimoto--Nishimura and Dinh--Forcella showed that radial solutions with negative energy must either blow up in finite time or exist globally while their $H^1$-norm grows without bound. In this paper, we prove that every radial $H^1\times H^1$ solution with negative energy blows up in finite time, both forward and backward in time. No finite-variance assumption is required. The main ingredient is a localized virial argument based on the bounded exponential weight \[ \nablaϕ_R(x)=2x e^{-|x|^2/R^2}. \] A radial weighted interpolation estimate allows us to control the nonlinear error terms by the corresponding weighted kinetic term, up to an $O(R^{-2})$ error depending only on the conserved mass. Moreover, the localized virial quantity itself can be bounded directly in terms of the same weighted kinetic defect. Combining these estimates yields a superlinear Riccati-type differential inequality, which cannot persist for all time and therefore forces finite-time blow-up.
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