  A semiconductor Boltzmann equation with a non-linear BGK-type collision
operator is analyzed for a cloud of ultracold atoms in an optical lattice:
  \[
  \partial_t f + \nabla_p\epsilon(p)\cdot\nabla_x f - \nabla_x n_f\cdot\nabla_p
f = n_f(1- n_f)(\mathcal{F}_f-f), \quad x\in\mathbb{R}^d, p\in\mathbb{T}^d,
t>0.
  \]
  This system contains an interaction potential
$n_f(x,t):=\int_{\mathbb{T}^d}f(x,p,t)dp$ being significantly more singular
than the Coulomb potential, which is used in the Vlasov-Poisson system. This
causes major structural difficulties in the analysis. Furthermore, $\epsilon(p)
= -\sum_{i=1}^d$ $\cos(2\pi p_i)$ is the dispersion relation and
$\mathcal{F}_f$ denotes the Fermi-Dirac equilibrium distribution, which depends
non-linearly on $f$ in this context.
  In a dilute plasma - without collisions (r.h.s$.=0$) - this system is closely
related to the Vlasov-Dirac-Benney equation. It is shown for analytic initial
data that the semiconductor Boltzmann equation possesses a local, analytic
solution. Here, we exploit the techniques of Mouhout and Villani by using
Gevrey-type norms which vary over time. In addition, it is proved that this
equation is locally ill-posed in Sobolev spaces close to some Fermi-Dirac
equilibrium distribution functions.
