  We consider the compressible barotropic Navier-Stokes equations in a
half-line and study the time-asymptotic behavior toward the outgoing viscous
shock wave. Precisely, we consider the two boundary problems: impermeable wall
and inflow problems, where the velocity at the boundary is given as a constant
state. For both problems, when the asymptotic profile determined by the
prescribed constant states at the boundary and far-fields is a viscous shock,
we show that the solution asymptotically converges to the shifted viscous shock
profiles uniformly in space, under the condition that initial perturbation is
small enough in H1 norm. We do not impose the zero mass condition on initial
data, which improves the previous results by Matsumura and Mei [20] for
impermeable case, and by Huang, Matsumura and Shi [8] for inflow case.
Moreover, for the inflow case, we remove the assumption in [8]. Our results are
based on the method of a-contraction with shifts, as the first extension of the
method to the boundary value problems.
