Boltzmann equations with quantum effects (1): Long time behavior of spatial decay solutions

The Boltzmann equations for Fermi-Dirac particles and Bose-Einstein particles, both in the absence of external force fields, are combined into a more general form called the Boltzmann equation with quantum effects (BQE). It is assumed that the initial data f(x, v, 0) satisfies 0 ≤ f(x, v, 0) < cΦ(x, v, 0) for a positive constant c and certain types of control functions Φ(x, v, t). Then within a given function space B(Φ), we prove that f(x + tv, v, t) uniformly converges to f = (x, v) in a certain norm where f = (x, v) = limf(x + tv, v, t) and different initial data determines different long time limits.