Global existence of small amplitude solutions to one-dimensional nonlinear Klein-Gordon systems with different masses

We study the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the rate $O(t^{-(1/2-1/p)})$ in $L^p$, $p\in[2,\infty]$ as $t$ tends to infinity even in the case of mass resonance, if the Cauchy data are sufficiently small, smooth and compactly supported.