By assuming certain local energy estimates on (1+3)-dimensional asymptotically flat space-time, we study the existence portion of the Strauss type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in [Metcalfe-Tataru 2012]. These estimates can be used to prove a series of weighted Strichartz and KSS type estimates, for wave equations on asymptotically flat space-time. Then we apply the spacetime estimates to obtain the lower bound of the lifespan when the nonlinear exponents p and q ≥ 2. In particular, our bound for the subcritical case is sharp in general and we extend the known region of (p, q) to admit global solutions. In addition, the initial data are not required to be compactly supported, when p, q > 2.