Stability of a coupled wave-Klein-Gordon system with non-compactly supported initial data

In this paper we study global nonlinear stability for a system of semilinear wave and Klein- Gordon equations with quadratic nonlinearities. We consider nonlinearities of the type of wave-Klein-Gordon interactions where there are no derivatives on the wave component. The initial data are assumed to have a suitable polynomial decay at infinity, but are not necessarily compactly supported. We prove small data global existence, sharp pointwise decay estimates and linear scattering results for the solutions. We give a sufficient condition which combines the Sobolev norms of the nonlinearities on flat time slices outside the light cone and the norms of them on truncated hyperboloids inside the light cone, for the linear scattering of the solutions.