Global existence of small amplitude solutions to nonlinear coupled wave-Klein-Gordon systems in four space-time dimension with hyperboloidal foliation method

In this article one will develop a new type of energy method based on a foliation of spacetime into hyperboloidal hypersurfaces . As we will see, with this method, some classical results such as global existence and almost global existence of regular solutions to the quasi-linear wave equations and Klein-Gordon equations will be established in a much simpler and much more natural way. Most importantly, the global existence of regular solutions to a general type of coupled quasilinear wave-Klein-Gordon system will be established. All of this suggests that compared with the classical method, this hyperboloidal foliation of space-time may be a more natural way to regard the wave operator.