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

Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions