  For the physical vacuum free boundary problem with the sound speed being
$C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the
three-dimensional compressible Euler equations with damping, the global
existence of spherically symmetric smooth solutions is proved, which are shown
to converge to Barenblatt self-similar solutions of the porous media equation
with the same total masses when initial data are small perturbations of
Barenblatt solutions. The pointwise convergence with a rate of density, the
convergence rate of velocity in supreme norm and the precise expanding rate of
physical vacuum boundaries are also given by constructing nonlinear functionals
with space-time weights featuring the behavior of solutions in large time and
near the vacuum boundary and the center of symmetry, the nonlinear energy
estimates and elliptic estimates.
