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

Abstract

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