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

We consider coupled systems of nonlinear wave and Klein–Gordon equations in two space dimensions with cubic nonlinearities. For these kinds 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.