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This article is devoted to the mass-less energy critical Maxwell-Klein-Gordonsystem in 4+1 dimensions. In earlier work of the second author, joint withKrieger and Sterbenz, we have proved that this problem has globalwell-posedness and scattering in the Coulomb gauge for small initial data. Thisarticle is the second of a sequence of three papers of the authors, whose goalis to show that the same result holds for data with arbitrarily large energy.Our aim here is to show that large data solutions persist for as long as onehas small energy dispersion; hence failure of global well-posedness must beaccompanied with a non-trivial energy dispersion.

Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation