A. Komech et al., LONG-TIME ASYMPTOTICS FOR A CLASSICAL PARTICLE INTERACTING WITH A SCALAR WAVE-FIELD, Communications in partial differential equations, 22(1-2), 1997, pp. 307-335
We consider the Hamiltonian system consisting of scalar wave held and
a single particle coupled in a translation invariant manner. The point
particle is subject to a confining external potential. The stationary
solutions of the system are a Coulomb type wave field centered at tho
se particle positions for which the external force vanishes. We prove
that solutions of finite energy converge, in suitable local energy sem
inorms, to the set of stationary solutions in the long time limit t --
> +/-infinity. The rate of relaxation to a stable stationary solution
is determined by spatial decay of initial data.