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We study in $\mathbb{R}^{3+1}$ a system of nonlinearly coupled Klein-Gordonequations under null condition, with (possibly vanishing) mass varying in theinterval $[0, 1]$. Our goal is three folds: 1) we want to establish the globalwell-posedness result to the system which is uniform in terms of the massparameter; 2) we want to obtain unified pointwise decay result for the solutionto the system, in the sense that the solution decays more like a wave component(independent of the mass parameter) in certain range of time, while thesolution decays as a Klein-Gordon component with a factor depending on the massparameter in the other part of the time range; 3) the solution to theKlein-Gordon system converges to the solution to the corresponding wave systemin certain sense when the mass parameter goes to 0. In order to achieve thesegoals, we will rely on both the flat and the hyperboloidal foliation of thespacetime.

The zero mass problem for Klein-Gordon equations: quadratic null interactions