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In this paper, we are concerned with the Cauchy problem for one-dimensionalcompressible isentropic Navier-Stokes equations with density-dependentviscosity $\mu(\rho)=\rho^\alpha (\alpha>0)$ and pressure$P(\rho)=\rho^{\gamma}\ (\gamma>1)$. We will establish the global existence andasymptotic behavior of weak solutions for any $\alpha>0$ and $\gamma>1$ underthe assumption that the density function keeps a constant state at far fields.This enlarges the ranges of $\alpha$ and $\gamma$ and improves the previousresults presented by Jiu and Xin. As a result, in the case that$0<\alpha<\frac12$, we obtain the large time behavior of the strong solutionobtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

On One-dimensional Compressible Navier-Stokes Equations with Degenerate Viscosity and Constant State at Far fields