  The thermalization of a gas towards a Maxwellian velocity distribution with
the background temperature is described by a kinetic relaxation model. The sum
of the kinetic energy of the gas and the thermal energy of the background are
conserved, and the heat flow in the background is governed by the Fourier law.
  For the coupled nonlinear system of the kinetic and the heat equation,
existence of solutions is proved on the one-dimensional torus. Spectral
stability of the equilibrium is shown on the torus in arbitrary dimensions by
hypocoercivity methods. The macroscopic limit towards a nonlinear
cross-diffusion problem is carried out formally.
