Time decay of solutions of coupled Maxwell-Klein-Gordon equations

Abstract

We obtain the optimal time decay of the solutions of the coupled Maxwell-Klein-Gordon equations in four dimensional spacetime, provided the initial data are what we define as Coulomb. In other words, the initial data are such that the Klein-Gordon field is smooth and compactly supported and the Maxwell field is electrostatic outside this support. The problem involves charge, therefore, the initial data do not satisfy either fast decay or any smallness condition. In spite of that, we are able to obtain our result using the inversion map of the lightcone of a carefully selected origin. We thus, avoided the blow-up that takes place when using the usual conformal transformation to the Einstein spacetime.