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 α ∈ [0, 1], and the linear interaction with the reservoirs by (1 − α), we prove that for some α close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is unique, and there are no spatially non-uniform NESS with a spatial density ρ belonging to L p for any p > 1. We also show that for all α ∈ [0, 1], the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.
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