  In this paper, we consider the existence and asymptotic behavior of the
fermionic quantum BGK model, which is a relaxation model of the quantum
Boltzmann equation for fermions. More precisely, we establish the existence of
unique classical solutions and their exponentially fast stabilization when the
initial data starts sufficiently close to a global Fermi-Dirac distribution. A
key difficulty unobserved in the study of the classical BGK model is that we
must verify that the equilibrium parameters is uniquely determined through a
set of nonlinear equations in each iteration step.
