  We study the non-linear BGK model in 1d coupled to a spatially varying
thermostat. We show existence, local uniqueness and linear stability of a
steady state when the linear coupling term is large compared to the non-linear
self interaction term. This model possesses a non-explicit spatially dependent
non-equilibrium steady state. We are able to successfully use hypocoercivity
theory in this case to prove that the linearised operator around this steady
state posesses a spectral gap.
