Uniqueness of the non-equilibrium steady state for a 1d BGK model in kinetic theory

We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter alpha is an element of[0,1], and the linear interaction with the reservoirs by (1-alpha), we prove that for some alpha close enough to zero, the explicit spatially uniform non-equilibrium steady state (NESS) is unique, and there are no spatially non-uniform NESS with a spatial density rho belonging to Lp for any p>1. We also show that for all alpha is an element of[0,1], the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.