Asymptotics and Scattering for Massive Maxwell–Klein–Gordon Equations

We study the asymptotic behavior and the scattering from infinity problem for the massive Maxwell–Klein–Gordon system in the Lorenz gauge, which were previously only studied for the massless system. For a general class of initial data, in particular of nonzero charge, we derive the precise asymptotic behaviors of the solution, where we get a logarithmic phase correction for the complex Klein–Gordon field, and a combination of interior homogeneous field, radiation fields, and an exterior charge part for the gauge potentials. Moreover, we also derive a formula for the charge at infinite time, which shows that the charge is concentrated at timelike infinity, a phenomenon drastically different from the massless case. After deriving the forward asymptotics, we formulate the scattering from infinity problem by defining the correct notion of scattering data, and then solve this problem. We show that one can determine the correct charge contribution using the information at timelike infinity, which is a crucial step for us to obtain backward solutions not only for the reduced equations in the Lorenz gauge but also for the original physical system.