  This paper is a continuation of the works in \cite{Euler} and \cite{NS},
where the authors have established the global existence of smooth compressible
flows in infinitely expanding balls for inviscid gases and viscid gases,
respectively. In this paper, we are concerned with the global existence and
large time behavior of compressible Boltzmann gases in an infinitely expanding
ball. Such a problem is one of the interesting models in studying the theory of
global smooth solutions to multidimensional compressible gases with time
dependent boundaries and vacuum states at infinite time. Due to the
conservation of mass, the fluid in the expanding ball becomes rarefied and
eventually tends to a vacuum state meanwhile there are no appearances of vacuum
domains in any part of the expansive ball, which is easily observed in finite
time. In the present paper, we will confirm this physical phenomenon for the
Boltzmann equation by obtaining the exact lower and upper bound on the
macroscopic density function.
