  We consider the global existence and large-time asymptotic behavior of strong
solutions to the Cauchy problem of the three-dimensional nonhomogeneous
incompressible Navier-Stokes equations with density-dependent viscosity and
vacuum. We establish some key a priori exponential decay-in-time rates of the
strong solutions. Then after using these estimates, we also obtain the global
existence of strong solutions in the whole three-dimensional space, provided
that the initial velocity is suitably small in the $\dot H^\beta$-norm for some
$\beta\in(1/2,1].$ Note that this result is proved without any smallness
conditions on the initial density. Moreover, the density can contain vacuum
states and even have compact support initially.
