The Kompaneets equation governs the evolution of a photon energy spectrum due to Compton scattering in a spatially homogeneous plasma. We prove some 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 a photon loss occurs, and show that once a loss is initiated then it persists forever. We prove that every solution has a large-time limit that is a Bose-Einstein density that can be characterized in terms of the total photon loss. Additionally, we provide some 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 an L^1 contraction. None of these results impose a boundary condition at the zero-energy boundary.
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