On the Cauchy problem of Boltzmann equation with a very soft potential

This work proves the global existence to Boltzmann equation in whole space with very soft potential $$\gamma \in [0,d)$$ and angular cutoff, framework of small perturbation equilibrium state. In this article, we generalize estimate on linearized collision operator L case obtain spectrum structure correspondingly. The classic solution can be derived by method strongly continuous semigroup. For potential, could not give spectral gap; hence, have consider a weighted velocity order algebraic decay time.