Asymptotic behavior of small solutions to the Vlasov--Klein--Gordon system in high dimensions

We study the asymptotic behavior of small solutions to the Vlasov–Klein–Gordon system in high dimensions. The standard argument of Glassey and Strauss \cite{GS87} for studying small solutions to the Vlasov–Maxwell system does not apply to the Vlasov–Klein–Gordon system due to the massiveness of the Klein–Gordon field. In this paper we use the vector field method and consider solutions in dimensions $n\ge 4$ with the hyperboloidal foliation of the Minkowski spacetime to obtain the asymptotic properties for the Vlasov–Klein–Gordon system.