  Jeans instability is analysed in an expanding universe within the framework
of BGK model of the Boltzmann equation and Poisson equations. The background is
characterized by a comoving Maxwellian distribution function and a space-time
Newtonian gravitational potential which satisfy the BGK model of the Boltzmann
and Poisson equations without the necessity to invoke "Jeans swindle". The
perturbations of the distribution function and Newtonian gravitational
potentials from their background states are represented by plane waves of small
amplitudes and a differential equation for the density contrast is determined.
The density contrast differential equation was solved numerically and it is
shown: (i) Jeans instability is characterized by perturbation wavelengths
larger than Jeans wavelength where the density contrast grows with time. The
growth of the density contrast is less accentuated for the case where the
particle collisions are considered due to an energy dissipation; (ii) for
perturbation wavelengths smaller than Jeans wavelength the density contrast has
an oscillatory behavior in time and the oscillations for the case where the
collisions are taken into account fade away in time due to the energy
dissipation.
