Global existence for systems of nonlinear wave and Klein-Gordon equations in two space dimensions under a kind of the weak null condition

TL;DR

This paper proves the global existence of small data solutions for coupled nonlinear wave and Klein-Gordon equations in two dimensions under a weak null condition, extending previous results that required a null condition.

Contribution

It introduces a new condition related to the weak null condition ensuring global existence for these systems, broadening the class of nonlinearities covered.

Findings

Global existence of solutions under weak null condition

Remarks on asymptotic behavior of solutions

Extension beyond classical null condition results

Abstract

We consider the coupled systems of nonlinear wave and Klein-Gordon equations in two space dimensions with cubic nonlinearity. For this kind of systems, the small data global existence is already known if the cubic nonlinearity satisfies a certain condition related to the null condition. In this article, our aim is to investigate the small data global existence under a condition related to the weak null condition. We also make a remark on the asymptotic behavior of global solutions.