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With the existence and uniqueness of a vector value solution for the fullnon-linear homogeneous Boltzmann system of equations describing multi-componentmonatomic gas mixtures for binary interactions proved [8], we present in thismanuscript several properties for such a solution. We start by proving the gainof integrability of the gain term of the multispecies collision operator,extending the work done previously for the single species case [3]. Inaddition, we study the integrability properties of the multispecies collisionoperator as a bilinear form, revisiting and expanding the work done for asingle gas [2]. With these estimates, together with a control by below for theloss term of the collision operator as in [8], we develop the propagation forthe polynomially and exponentially $\beta$-weighted $L^p_\beta$-norms for thevector value solution. Finally, we extend such $L^p_\beta$-norms propagationproperty to $p=\infty$.

Propagation of $L^p_{\beta}$-norm, $ 1