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arXiv 2004.10467math.AP
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Klein-Gordon方程的零质量问题:二次零相互作用

The zero mass problem for Klein-Gordon equations: quadratic null interactions

Shijie Dong

摘要:

我们在$\mathbb{R}^{3+1}$中研究了零条件下的非线性耦合Klein-Gordon方程组,其质量(可能为零)在区间$[0, 1]$中变化。我们的目标有三个方面:1)对于质量参数一致的系统,我们要建立其整体适定性结果; 2)我们希望得到系统解的统一的逐点衰减结果,即解的衰减更像波分量(与质量参数无关)在一定时间范围内,而在另一段时间内,溶液作为Klein-Gordon分量衰减,衰减因子取决于质量参数; 3)当质量参数为0时,Klein-Gordon系统的解在一定意义下收敛于相应波系的解。为了实现这些目标,我们将依赖于时空的平坦叶理和双曲面叶理。

英文摘要:

We study in $\mathbb{R}^{3+1}$ a system of nonlinearly coupled Klein-Gordon equations under null condition, with (possibly vanishing) mass varying in the interval $[0, 1]$. Our goal is three folds: 1) we want to establish the global well-posedness result to the system which is uniform in terms of the mass parameter; 2) we want to obtain unified pointwise decay result for the solution to the system, in the sense that the solution decays more like a wave component (independent of the mass parameter) in certain range of time, while the solution decays as a Klein-Gordon component with a factor depending on the mass parameter in the other part of the time range; 3) the solution to the Klein-Gordon system converges to the solution to the corresponding wave system in certain sense when the mass parameter goes to 0. In order to achieve these goals, we will rely on both the flat and the hyperboloidal foliation of the spacetime.

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