Large Time Behavior of the Klein-Gordon-Schr\"{o}dinger system

TL;DR

This paper proves global existence and scattering for small localized solutions of a coupled Klein-Gordon-Schrödinger system in three dimensions, using space-time resonance methods to handle quadratic nonlinearities and resonant sets.

Contribution

It introduces a novel application of space-time resonance analysis to a coupled Klein-Gordon-Schrödinger system with quadratic nonlinearities in three dimensions.

Findings

Established global existence for small solutions.

Proved scattering behavior in three dimensions.

Handled complex resonant interactions without null structures.

Abstract

We establish the global existence and scattering for small and localized solutions of the Klein-Gordon-Schr\"{o}dinger system in three dimensions. The system consists of coupled semilinear Schr\"{o}dinger and Klein-Gordon equations with quadratic nonlinearities. This model is motivated by the study of plasma oscillations arising from the Hartree equation near a translation-invariant equilibrium with the Coulomb potential. Our proof relies on the space-time resonance method. The main difficulty comes from the two dimensional space-time resonant set and the absence of null form structure.