Nonlinear wave equation in a cosmological Kaluza Klein spacetime

We prove the global existence for the Cauchy problem of a semi-linear wave equation with an arbitrary quadratic form in a cosmological Kaluza-Klein spacetime, which is the product of a Milne universe with S-1. The data prescribed on t = t(0) are small, and t(0) > 0 can arbitrarily approach the t = 0 singularity. Our proof relies on a decomposition of the wave equation into the zero and non-zero modes, and a crucial nonlinear structure after this decomposition is observed. In addition, we introduce various modified energies and the associated energy identities according to different expectations of decaying estimates.