  We analyze systems of semilinear wave equations in $3+1$ dimensions whose
associated asymptotic equation admit bounded solutions for suitably small
choices of initial data. Under this special case of the weak null condition,
which we refer to as the \textit{bounded weak null condition}, we prove the
existence of solutions to these systems of wave equations on neighborhoods of
spatial infinity under a small initial data assumption. Existence is
established using the Fuchsian method. This method involves transforming the
wave equations into a Fuchsian equation defined on a bounded spacetime region.
The existence of solutions to the Fuchsian equation then follows from an
application of the existence theory developed in \cite{BOOS:2020}. This, in
turn, yields, by construction, solutions to the original system of wave
equations on a neighborhood of spatial infinity.
