  The nonhomogeneous Navier-Stokes equations with density-dependent viscosity
is studied in three-dimensional (3D) exterior domains with nonslip or slip
boundary conditions. We prove that the strong solutions exists globally in time
provided that the gradient of the initial velocity is suitably small. Here the
initial density is allowed to contain vacuum states. Moreover, after developing
some new techniques and methods, the large-time behavior of the strong
solutions with exponential decay-in-time rates is also obtained.
