We study long-time dynamics in a simplified Kompaneets equation. The Kompaneets equation describes evolution of photon energy spectrum due to Compton scattering of photons by electrons, an important energy transport mechanism in certain plasmas. For our model, we prove global existence for initial data with any finite moment, convergence to equilibrium in large time, and failure to conserve photon number for large solutions, due to formation of a shock at the zero-energy domain boundary. This is joint work with Dave Levermore and Hailiang Liu.