Nordström引力理论中非线性wave equation的整体存在性
Global existence of a nonlinear wave equation arising from Nordström's theory of gravitation
摘要：
我们证明了具有源项和宇宙常数的非线性Nordström理论的整体经典解的存在性，其中假设源项在适当的范数下是小的，而在某些情况下不需要对初始数据做小的假设.在这个理论中，引力场被描述为一个单一的标量函数，满足一定的半线性波动方程。我们考虑空间周期性偏离背景度规，这就是为什么我们在索伯列夫空间$H^m(\setT^3)$中研究三维环面$\setT^3$上的半线性波动方程。我们采用两种方法来获得整体解的存在性，第一种方法是利用傅里叶级数，第二种方法是将半线性波动方程非常规地写成对称双曲方程组.我们还提供了有关这些解决方案的渐近行为的结果，最后，爆破的结果，如果我们的整体存在定理的条件不满足。
英文摘要：
We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus $\setT^3$ in the Sobolev spaces $H^m(\setT^3)$. We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up result if the conditions of our global existence theorems are not met.
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