  We study the evolution of electrostatic plasma waves, using the relativistic
Vlasov equation extended by the Landau-Lifshitz radiation reaction, accounting
for the back-reaction due to the emission of single particle Larmor radiation.
In particular, the Langmuir wave damping is calculated as a function of
wavenumber, initial temperature, and initial electric field amplitude.
Moreover, the background distribution function loses energy in the process, and
we calculate the cooling rate as a function of initial temperature and initial
wave amplitude. Finally, we investigate how the relative magnitude of wave
damping and background cooling varies with the initial parameters. In
particular, it is found that the relative contribution to the energy loss
associated with background cooling decreases slowly with the initial wave
amplitude.
