  In the paper, assuming that the motion of rarefied gases in a bounded domain
is governed by the angular cutoff Boltzmann equation with diffuse reflection
boundary, we study the effects of both soft intermolecular interaction and
non-isothermal wall temperature upon the long-time dynamics of solutions to the
corresponding initial boundary value problem. Specifically, we are devoted to
proving the existence and dynamical stability of stationary solutions whenever
the boundary temperature has suitably small variations around a positive
constant. For the proof of existence, we introduce a new mild formulation of
solutions to the steady boundary-value problem along the speeded backward
bicharacteristic, and develop the uniform estimates on approximate solutions in
both $L^2$ and $L^\infty$. Such mild formulation proves to be useful for
treating the steady problem with soft potentials even over unbounded domains.
In showing the dynamical stability, a new point is that we can obtain the
sub-exponential time-decay rate in $L^\infty$ without losing any velocity
weight, which is actually quite different from the classical results as in
[11,43] for the torus domain and essentially due to the diffuse reflection
boundary and the boundedness of the domain.
