Global dynamics of the Kompaneets equation

The Kompaneets equation describes energy transport in low-density (or
high temperature) plasmas where the dominant energy exchange mechanism
is Compton scattering. The equation itself is a one dimensional
non-linear parabolic equation with a diffusion coefficient that vanishes
at the boundary. This degeneracy, combined with the nonlinearity causes
an out-flux of photons with zero energy, often interpreted as a
Bose--Einstein condensate. This talk will describe several results about
the long time behavior of two models of these these equations including
convergence to equilibrium, persistence of the condensate, sufficient
conditions under which it forms, sufficient conditions under which it
doesn't form and a loss formula for the mass of the condensate. I will
also talk about some ongoing work that obtains most of these results for
the full Kompaneets equation.