  We study the coupled wave-Klein-Gordon systems, introduced by LeFloch-Ma and
then Ionescu-Pausader, to model the nonlinear effects from the
Einstein-Klein-Gordon equation in harmonic coordinates. We first go over a
slightly simplified version of global existence based on LeFloch-Ma, and then
derive the asymptotic behavior of the system. The asymptotics of the
Klein-Gordon field consist of a modified phase times a homogeneous function,
and the asymptotics of the wave equation consist of a radiation field in the
wave zone and an interior homogeneous solution coupled to the Klein-Gordon
asymptotics. We then consider the inverse problem, the scattering from
infinity. We show that given the type of asymptotic behavior at infinity, there
exist solutions of the system that present the exact same behavior.

Analysis of PDEs