The Maxwell-Lorentz system of a rigid charge

We prove global existence and uniqueness of classical solutions for the Maxwell-Lorentz system of a nonrotating rigid charge distribution, i.e. the relativistic dynamics of a nonrotating extended electron, which is subject to its own electromagnetic fields and an external potential. Local existence and uniqueness is achieved via the contraction mapping principle. Suitable a-priori-bounds yield global existence. We show that in case of a negative bare mass and an attracting external potential the stationary solution is unstable. We believe that this result clarifies the origin of the so-called "runaway"-solutions, which appear when the limit to a point charge is taken, formally described by the so called Lorentz-Dirac equation for the radiating electron.