For the physical vacuum free boundary problem with the sound speed being
C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible
Euler equations with damping, the global existence of the smooth solution
is proved, which is shown to converge to the Barenblatt self-similar solution
for the porous media equation with the same total mass when the initial datum is
a small perturbation of the Barenblatt solution. The pointwise convergence with
a rate of density, the convergence rate of velocity in the supremum norm, and the
precise expanding rate of the physical vacuum boundaries are also given. The
proof is based on a construction of higher-order weighted functionals with both
space and time weights capturing the behavior of solutions both near vacuum
states and in large time, an introduction of a new ansatz, higher-order nonlinear
energy estimates, and elliptic estimates.