Cauchy problem for NLKG in modulation spaces with noninteger powers

In this paper, we consider the Cauchy problem for the nonlinear Klein-Gordon equation whose nonlinearity is $\mathrm{\mid }u{\mathrm{\mid }}^{k}u$ in the modulation space, where $k$ is not an integer. Our method can be applied to other equations whose nonlinear parts have regularity estimates. We also study the global solution with small initial value for the Klein-Gordon-Hartree equation. By this we can show some advantages of modulation spaces both in high and low regularity cases.