  We study the equilibrating effects of the boundary and intermolecular
collision in the kinetic theory for rarefied gases. We consider the
Maxwell-type boundary condition, which has weaker equilibrating effect than the
commonly studied diffuse reflection boundary condition. The gas region is the
spherical domain in $\mathbb{R}^d$, $d=1,2$. First, without the equilibrating
effect of collision, we obtain the algebraic convergence rates to the steady
state of free molecular flow with variable boundary temperature. The
convergence behavior has intricate dependence on the accommodation coefficient
of the Maxwell-type boundary condition. Then we couple the boundary effect with
the intermolecular collision and study their interaction. We are able to
construct the steady state solutions of the full Boltzmann equation for large
Knudsen numbers and small boundary temperature variation. We also establish the
nonlinear stability with exponential rate of the stationary Boltzmann
solutions. Our analysis is based on the explicit formulations of the boundary
condition for symmetric domains.
