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We study the degenerate linear Boltzmann equation inside a bounded domainwith the Maxwell and the Cercignani-Lampis boundary conditions, twogeneralizations of the diffuse reflection, with variable temperature. Thisincludes a model of relaxation towards a space-dependent steady state. For bothboundary conditions, we prove for the first time the existence of a steadystate and a rate of convergence towards it without assumptions on thetemperature variations. Our results for the Cercignani-Lampis boundarycondition make also no hypotheses on the accommodation coefficients. The provenrate is exponential when a control condition on the degeneracy of the collisionoperator is satisfied, and only polynomial when this assumption is not met, inline with our previous results regarding the free-transport equation. We alsoprovide a precise description of the different convergence rates, includinglower bounds, when the steady state is bounded. Our method yields constructiveconstants.

Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions