We prove the global existence for the Cauchy problem of a semi-linear wave equation with an arbitrary quadratic form in a cosmological Kaluza–Klein spacetime, which is the product of a Milne universe with S¹. The data prescribed on t = t₀ are small, and t₀ > 0 can arbitrarily approach the t = 0 singularity. Our proof relies on a decomposition of the wave equation into the zero and non-zero modes, and a crucial nonlinear structure after this decomposition is observed. In addition, we introduce various modified energies and the associated energy identities according to different expectations of decaying estimates.

中文翻译：

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宇宙学 Kaluza Klein 时空中的非线性波动方程

我们证明了宇宙学 Kaluza-Klein 时空中具有任意二次型的半线性波动方程的柯西问题的全局存在性，该方程是 Milne 宇宙与S ¹的乘积。t = t ₀上规定的数据很小，t ₀ > 0 可以任意逼近t = 0 奇点。我们的证明依赖于将波动方程分解为零和非零模式，并观察到这种分解后的关键非线性结构。此外，我们根据衰减估计的不同期望引入了各种修改的能量和相关的能量恒等式。