  This paper is concerned with the boundary-value problem on the Boltzmann
equation in bounded domains with diffuse-reflection boundary where the boundary
temperature is time-periodic. We establish the existence of time-periodic
solutions with the same period for both hard and soft potentials, provided that
the time-periodic boundary temperature is sufficiently close to a stationary
one which has small variations around a positive constant. The dynamical
stability of time-periodic profiles is also proved under small perturbations,
and this in turn yields the non-negativity of the profile. For the proof, we
develop new estimates in the time-periodic setting.
