High-velocity tails of the inelastic and the multi-species mixture Boltzmann equations

We study high-velocity tails of some homogeneous Boltzmann equations on $v\in {\mathbb{R}}_v^d$. First, we consider spatially homogeneous inelastic Boltzmann equation with noncutoff collision kernel, in the case of moderately soft potentials. We also study spatially homogeneous mixture Boltzmann equations : for both noncutoff collision kernel with moderately soft potentials and cutoff collision kernel with hard potentials. In the case of noncutoff inelastic Boltzmann, we obtain $$f(t,v)\ge a(t)e^{-b(t)\mathrm{\mid }v{\mathrm{\mid }}^p},2<p<6.213$$ by extending Cancellation lemma and spreading lemma and assuming $f\in C^{\mathrm{\infty }}$. For the Mixture type Boltzmann equations, we prove Maxwellian $p=2$.