The Lorentz-Dirac equation, I

The Lorentz-Dirac equation (LDE) tau x''' = x " = V(x) models the point lim it of the Maxwell-Lorentz equation describing the interaction of a charged extended particle with the electromagnetic field. Since (LDE) admits soluti ons which accelerate even if they are outside the zone of interaction, Dira c proposed to study so-called "non runaway" solutions satisfying the condit ion x "(t) --> 0 as t --> +infinity. We study the scattering of particles f or a localized potential barrier V(x). We show, using global bifurcation te chniques, that for every T > T-0 there exists a reflection solution with "r eturning time" T, and for every T > 0 there exists a transmission solution with "transmission time" T. Furthermore, some qualitative properties of the solutions are proved; in particular, it is shown that for increasing T, th ese solutions spend more and more time near the maximum point s(0) of V.