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We consider a class of nonlinear Boltzmann equations describing return tothermal equilibrium in a gas of colliding particles suspended in a thermalmedium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times\mathbb{T}^3).$ Special solutions of these equations, called "Maxwellians," arespatially homogenous static Maxwell velocity distributions at the temperatureof the medium. We prove that, for dilute gases, the solutions corresponding tosmooth initial conditions in a weighted $L^{1}$-space converge to a Maxwellianin $L^{1},$ exponentially fast in time.

Exponential Convergence to the Maxwell Distribution For Some Class of Boltzmann Equations