  By assuming certain local energy estimates on $(1+3)$-dimensional
asymptotically flat space-time, we study the existence portion of the
\emph{Strauss} type wave system. Firstly we give a kind of space-time estimates
which are related to the local energy norm that appeared in \cite{MR2944027}.
These estimates can be used to prove a series of weighted \emph{Strichartz} and
\emph{KSS} type estimates, for wave equations on asymptotically flat
space-time. Then we apply the space-time estimates to obtain the lower bound of
the lifespan when the nonlinear exponents $p$ and $q\ge 2$. 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 $p, q>2$.

Analysis of PDEs