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We prove global existence backwards from the scattering data posed atinfinity for the Maxwell Klein Gordon equations in Lorenz gauge satisfying theweak null condition. The asymptotics of the solutions to the Maxwell KleinGordon equations in Lorenz gauge were shown to be wave like at null infinityand homogeneous towards timelike infinity in arXiv:1803.11086 and expressed interms of radiation fields, and thus our scattering data will be given in theform of radiation fields in the backward problem. We give a refinement of theasymptotics results in arXiv:1803.11086, and then making use of thisrefinement, we find a global solution which attains the prescribed scatteringdata at infinity. Our work starts from the approach in [22] and is moredelicate since it involves nonlinearities with fewer derivatives. Our resultcorresponds to "existence of scattering states" in the scattering theory. Themethod of proof relies on a suitable construction of the approximate solutionfrom the scattering data, a weighted conformal Morawetz energy estimate and aspacetime version of Hardy inequality.

Scattering from Infinity of the Maxwell Klein Gordon Equations in Lorenz Gauge