  We prove the global existence and uniqueness of smooth solutions to the
one-dimensional barotropic Navier-Stokes system with degenerate viscosity
$\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possibly
two different far-fields, and the initial density remains positive globally in
time, for the initial data satisfying the same conditions. In addition, our
result works for any $\alpha>0$, i.e., for a large class of degenerate
viscosities. In particular, our models include the viscous shallow water
equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto
\cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where
smooth solutions connect possibly two different limits at the infinity on the
whole space.
