On the discrete Boltzmann equation with linear and nonlinear terms

For the discrete Boltzmann equation with linear and nonlinear terms, we show a boundedness of solutions with an explicit estimate and their asymptotic behavior when the momentum is conserved. Secondly when the Cauchy data are small, we show an exponential decay of solutions, for a model in which the nonlinear terms represent the `binary collisions\u27 and also `multiple collisions\u27