In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

在本文中，我们研究了二维空间中具有规范不变二次非线性的非线性Klein-Gordon方程复值解的大时性。为了找到可能的渐近行为，我们考虑了最终值问题。事实证明，一种可能的行为是与实值情况一样具有对数相位校正的线性解。但是，对数校正项的形状还有一个参数，该参数也由最终数据给出。在实际情况下，该参数是常量，因此无法看到其效果。但是，在复杂情况下，它通常会有所不同。还讨论了一维情况。