Asymptotic behavior of the solutions of maxwell—klein—gordon field equations in 4—dimensional minkowski space

. This paper studies the asymptotic behavior of the solutions of massive coupled Maxwell—Klein—Gordon field equations in the 4—dimensional Minkowski space for the case of small initial data with charge. The proof relies on gauge invariant energy estimates and geometric properties of the fields equations. The presence of charge together with the mass term in the Klein—Gordon equation are the novelties of this project and provide us with a situation which cannot be accomodated by standard methods such as the conformal transformation. A covariant Lie derivative operator for the Klein—Gordon field is introduced and allows us to handle the most troublesome terms in the error estimates.