problem of  has been proven in case of unbounded domains [@Pe89] and bounded domains [@Ri91; @PePu93], respectively. In case of bounded domains (such as $x\in\tilde{\mathbf{T}}^d$), these solutions are essentially bounded and unique [@PePu93]. For a space-inhomogeneous nonlinear BGK model with an external confinement potential, the global existence of solutions for its Cauchy problem and their strong convergence in $L^1$ to a Maxwellian equilibrium state has been proven recently [@BoCa09].

In the first part of this paper we shall study the linearization of the BGK equation around the centered Maxwellian with constant-in-$x$ temperature equal to one.

To this end we consider $f$ close to the global equilibrium $M_1(v)$, with $h$ defined by $f(x,v,t) = M_1(v)+h(x,v,t)$. Then $$\begin{aligned}
   \rho(x,t) &= 1 + \sigma(x,t) \qquad\text{with }\: \sigma(x,t):=\int_{{{\mathbb{R}}^d}} h(x,v,t) {\,\operatorname{d}\!v}\ , \nonumber \\
  (\rho u)(x,t) &= \int_{{{\mathbb{R}}^d}} vf(x,v,t) {\,\operatorname{d}\!v} =\mu(x,t) \qquad\text{with the vector function }\: \mu(x,t):=\int_{{{\mathbb{R}}^d}} vh(x,v,t) {\,\operatorname{d}\!v}\ ,\label{f:perturb} \\ 
  P(x,t) &= \frac1d \int_{{{\mathbb{R}}^d}} (|v-u|^2) f(x,v,t) {\,\operatorname{d}\!v} 
  = 1+\frac1d\big[\tau(x,t)-\frac{|\mu(x,t)|^2}{1+\sigma(x,t)}\big] \quad\text{with }\: \tau(x,t):={\int_{{\mathbb{R}}^d}}|v|^2 h(x,v,t) {\,\operatorname{d}\!v}\ . \nonumber $$ The conservation of the normalizations implies $$\label{mloc2}
\int_{\tilde{\mathbf{T}}^d} \sigma(x,t){\,\operatorname{d}\!\tilde x} =0\ ,  \qquad \int_{\tilde{\mathbf{T}}^d} \mu(x,t){\,\operatorname{d}\!\tilde x} =0 \ , \qquad{\rm and}\qquad   \int_{\tilde{\mathbf{T}}^d} \tau(x,t){\,\operatorname{d}\!\tilde x} =0\ .$$ The perturbation $h$ then satisfies $$\partial_t h (x,v,t) +  v\cdot \nabla_x 