On global behavior of solutions of the Maxwell-Klein-Gordon equations

TL;DR

This paper introduces a new method to analyze the long-term behavior of solutions to the Maxwell-Klein-Gordon equations in four-dimensional space, providing detailed decay estimates even with large charge.

Contribution

It presents the first comprehensive analysis of the global behavior and decay properties of large nonlinear charged scalar fields in the Maxwell-Klein-Gordon system.

Findings

Quantitative energy flux decay established

Integrated local energy decay proved

Hierarchy of r-weighted energy decay demonstrated

Abstract

It is known that the Maxwell-Klein-Gordon equations in $\mathbb{R}^{3+1}$ admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of $r$-weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields.