Asymptotic Behavior of the Solution to the Klein-Gordon-Zakharov Model in Dimension Two

TL;DR

This paper studies the long-term behavior of solutions to the Klein-Gordon-Zakharov equations in two spatial dimensions, establishing global existence for small data and analyzing their asymptotic decay using advanced geometric methods.

Contribution

It introduces a novel combination of hyperboloidal foliation and ghost weight techniques to prove small data global existence and asymptotic behavior for a challenging coupled wave and Klein-Gordon system.

Findings

Proved small data global existence for the Klein-Gordon-Zakharov system in 2D.

Established pointwise decay rates of solutions.

Extended analysis to certain quasilinear systems violating null conditions.

Abstract

Consider the Klein-Gordon-Zakharov equations in $\mathbb{R}^{1+2}$, and we are interested in establishing the small global solution to the equations and in investigating the pointwise asymptotic behavior of the solution. The Klein-Gordon-Zakharov equations can be regarded as a coupled semilinear wave and Klein-Gordon system with quadratic nonlinearities which do not satisfy the null conditions, and the fact that wave components and Klein-Gordon components do not decay sufficiently fast makes it harder to conduct the analysis. In order to conquer the difficulties, we will rely on the hyperboloidal foliation method and a minor variance of the ghost weight method. As a side result of the analysis, we are also able to show the small data global existence result for a class of quasilinear wave and Klein-Gordon system violating the null conditions.