  In this article one will develop a so-called hyperboloidal foliation method,
which is an energy method based on a foliation of space-time into hyperboloidal
hypersurfaces. This method permits to treat the wave equations and the
Klein-Gordon equations in the same framework so that one can apply it to the
coupled systems of wave and Klein-Gordon equations. As an application, one will
establish the global-in-time existence of small amplitude solution to the
coupled wave and Klei-Gordon equations with quadratic nonlinearity in four
space-time dimensions under certain conditions. Compared with those introduced
by S. Katayama, the conditions imposed in this article permit to include some
important nonlinear terms. All of these suggests that this method may be a more
natural way of regarding the wave operator.
