  We study the degenerate linear Boltzmann equation inside a bounded domain
with the Maxwell and the Cercignani-Lampis boundary conditions, two
generalizations of the diffuse reflection, with variable temperature. This
includes a model of relaxation towards a space-dependent steady state. For both
boundary conditions, we prove for the first time the existence of a steady
state and a rate of convergence towards it without assumptions on the
temperature variations. Our results for the Cercignani-Lampis boundary
condition make also no hypotheses on the accommodation coefficients. The proven
rate is exponential when a control condition on the degeneracy of the collision
operator is satisfied, and only polynomial when this assumption is not met, in
line with our previous results regarding the free-transport equation. We also
provide a precise description of the different convergence rates, including
lower bounds, when the steady state is bounded. Our method yields constructive
constants.
