AI 中文总结
本文研究Witten气泡时空上的半线性波动方程，在无对称性假设下证明小数据整体存在性，并对满足零条件的U(1)对称解改进衰减结果，关键贡献是线性波动方程的新估计。
AI 中文摘要
我们启动了关于Witten气泡时空（也称为“无物气泡”）上非线性波动方程的研究，该时空是$(4+1)$维爱因斯坦真空方程的一个$\text{SO}(3,1)\times \text{U}(1)$对称解。此时空由Witten引入，用以模拟Kaluza-Klein时空$\textbf{R}^{3+1}\times S^1$的半经典不稳定性（该时空在经典意义上是稳定的，由Huneau-Stingo-Wyatt的工作证明，见arXiv:2307.15267）。我们在无对称性假设下证明了小数据整体存在性结果，并对满足某种零条件的一类半线性方程的$\text{U}(1)$对称解改进了衰减结果。证明的关键在于对线性波动方程的新估计，该方程在物理学文献中由Bhawal和Viveshwara研究，在数学文献中由Bachelot研究（见arXiv:1601.03682）。
英文摘要
We initiate the study of nonlinear wave equations on the Witten bubble spacetime (also known as the "bubble of nothing"), which is an $\mathrm{SO}(3,1)\times \mathrm{U}(1)$-symmetric solution to the Einstein vacuum equations in $(4 + 1)$ dimensions. This spacetime was introduced by Witten to model the semiclassical instability of the Kaluza--Klein spacetime $\mathbf{R}^{3+1}\times S^1$ (which is classically stable by work of Huneau--Stingo--Wyatt (arXiv:2307.15267)). We prove a small-data global existence result without symmetry assumptions and an improved decay result for $\mathrm{U}(1)$-symmetric solutions to a class of semilinear equations satisfying a version of the null condition. Key to the proof are novel estimates for the linear wave equation, which has been studied in the physics literature by Bhawal and Viveshwara and in the mathematics literature by Bachelot (arXiv:1601.03682).
Comments57 pages, 2 figures
