Global Solution for Massive Maxwell-Klein-Gordon Equations

We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein-Gordon scalar field) in the exterior of the domain of influence of a compact set. This complements the previous well-known results, restricted to compactly supported initial conditions, based on the so-called hyperboloidal method. That method takes advantage of the commutation properties of the Maxwell and Klein-Gordon equations with the generators of the Poincaré group to resolve the difficulties caused by the fact that they have, separately, different asymptotic properties. Though the hyperboloidal method is very robust and applies well to other related systems, it has the well-known drawback of requiring compactly supported data. In this paper we remove this limitation based on a further extension of the vector field method adapted to the exterior region. Our method applies, in particular, to nontrivial charges. The full problem can then be treated by patching together the new estimates in the exterior with the hyperboloidal ones in the interior. This purely physical space approach introduced here maintains the robust properties of the old method and can thus be applied to other situations such as the coupled Einstein Klein-Gordon equation.