  In this paper we prove that the static solution of the Cauchy problem for a
massless real scalar field that is sourced by a point charge in $1+1$
dimensions is asymptotically stable under perturbation by compactly-supported
radiation. This behavior is due to the process of back-reaction. Taking the
approach of Kiessling, we rigorously derive the expression for the force on the
particle from the principle of total energy-momentum conservation. We provide a
simple, closed form for the particle's self-action, and show that it is
restorative in this model , i.e. proportional to negative velocity, and causes
the charge to return to rest after the radiation passes through. We establish
these results by studying the joint evolution problem for the particle-scalar
field system, and proving its global well-posedness and the claimed asymptotic
behavior.
