Asymptotics and Scattering for massive Maxwell-Klein-Gordon equations

TL;DR

This paper investigates the long-term behavior and scattering phenomena of the massive Maxwell-Klein-Gordon system in the Lorenz gauge, revealing unique charge concentration at timelike infinity and deriving precise asymptotic profiles for solutions.

Contribution

It extends the analysis of scattering and asymptotics from the massless to the massive Maxwell-Klein-Gordon system, including nonzero charge cases and charge concentration phenomena.

Findings

Logarithmic phase correction for the Klein-Gordon field

Charge concentrated at timelike infinity

Explicit asymptotic behavior and scattering solutions

Abstract

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, a combination of interior homogeneous function, radiation fields to null infinity and an exterior charge part for the gauge potentials. Moreover, we also derive a formula for 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 notion of the asymptotic profile, we prove the scattering from infinity by constructing backward solutions given the scattering data. We show that one can determine the correct charge contribution using the information at timelike infinity, which is important for us to obtain solutions not only for the reduced equations in the Lorenz gauge but also for the original physical system.