  The Kompaneets equation describes a field of photons exchanging energy by
Compton scattering with the free electrons of a homogeneous, isotropic,
non-relativistic, thermal plasma. This paper strives to advance our
understanding of how this equation captures the phenomenon of Bose-Einstein
condensation through the study of a model equation. For this model we prove
existence and uniqueness theorems for global weak solutions. In some cases a
Bose-Einstein condensate will form in finite time, and we show that it will
continue to gain photons forever afterwards. Moreover we show that every
solution approaches a stationary solution for large time. Key tools include a
universal super solution, a one-sided Oleinik type inequality, and an $L^1$
contraction.
