  The Kompaneets equation governs dynamics of the photon energy spectrum in
certain high temperature (or low density) plasmas. We prove several results
concerning the long-time convergence of solutions to Bose--Einstein equilibria
and the failure of photon conservation. In particular, we show the total photon
number can decrease with time via an outflux of photons at the zero-energy
boundary. The ensuing accumulation of photons at zero energy is analogous to
Bose--Einstein condensation. We provide two conditions that guarantee that
photon loss occurs, and show that once loss is initiated then it persists
forever. We prove that as $t\to \infty$, solutions necessarily converge to
equilibrium and we characterize the limit in terms of the total photon loss.
Additionally, we provide a few results concerning the behavior of the solution
near the zero-energy boundary, an Oleinik inequality, a comparison principle,
and show that the solution operator is a contraction in $L^1$. None of these
results impose a boundary condition at the zero-energy boundary.
