  Boundary effects are central to the dynamics of the dilute particles governed
by Boltzmann equation. In this paper, we study both the diffuse reflection and
the specular reflection boundary value problems for Boltzmann equation with
soft potential, in which the collision kernel is ruled by the inverse power
law. For the diffuse reflection boundary condition, based on an $L^2$ argument
and its interplay with intricate $L^\infty$ analysis for the linearized
Boltzmann equation, we first establish the global existence and then obtain the
exponential decay in $L^\infty$ space for the nonlinear Boltzmann equation in
general classes of bounded domain. It turns out that the zero lower bound of
the collision frequency and the singularity of the collision kernel lead to
some new difficulties for achieving the {\it a priori} $L^\infty$ estimates and
time decay rates of the solution. In the course of the proof, we capture some
new properties of the probability integrals along the stochastic cycles and
improve the $L^2-L^\infty$ theory to give a more direct approach to overcome
those difficulties. As to the specular reflection condition, our key
contribution is to develop a new time-velocity weighted $L^\infty$ theory so
that we could deal with the greater difficulties stemmed from the complicated
velocity relations among the specular cycles and the zero lower bound of the
collision frequency. From this new point, we are also able to prove the
solutions of the linearized Boltzmann equation tend to equilibrium
exponentially in $L^\infty$ space with the aid of the $L^2$ theory and a
bootstrap argument. These methods in the latter case can be applied to the
Boltzmann equation with soft potential for all other types of boundary
condition.
