Asymptotic Behavior of the Maxwell-Klein-Gordon system

In previous work on the Maxwell-Klein-Gordon system first existence and then decay estimates have been shown. Here we show that the Maxwell-Klein-Gordon in the Lorentz gauge satisfy the "weak null condition" and we give the detailed asymptotics for the scalar field and the potential. These asymptotics have two parts, one wave like along outgoing light cones at null infinity, and one homogeneous inside the light cone at time like infinity. Here the charge plays a crucial role in imposing an oscillating factor in the asymptotic system for the field, and in the null asymptotics for the potential. Similar results have previously been shown for Einstein's equations in wave coordinates, and the Maxwell-Klein-Gordon system apart from being interesting in itself also provides a simpler semilinear model of the quasilinear Enstein's equations.