Propagation of Moments and Sharp Convergence Rate for Inhomogeneous Noncutoff Boltzmann Equation with Soft Potentials

. We prove the well-posedness for the noncutoff Boltzmann equation with soft potentials when the initial datum is close to the global Maxwellian and has only polynomial decay at the large velocities in \ (L²\) space. As a result, we get the propagation of the exponential moments and the sharp rates of the convergence to the global Maxwellian which seems the first results for the original equation with soft potentials. The new ingredients of the proof lie in localized techniques, the semigroup method as well as the propagation of the polynomial and exponential moments in \ (L²\) space. Keywordsinhomogeneous Boltzmann equationsoft potentialsnon-cutoffpropagation of momentssharp convergence rateMSC codes35Q2035A9982C40