  In this paper, the smooth solution of the physical vacuum problem for the one
dimensional compressible Euler equations with time-dependent damping is
considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-H\"{o}lder
continuous. The coefficient of the damping depends on time, given by this form
$\frac{\mu}{(1+t)^\lambda}$, $\lambda$, $\mu>0$, which decays by order
$-\lambda$ in time. Under the assumption that $0<\lambda<1$, $0<\mu$ or
$\lambda=1$, $2<\mu$, we will prove the global existence of smooth solutions
and convergence to the modified Barenblatt solution of the related porous media
equation with time-dependent dissipation and the same total mass when the
initial data of the Euler equations is a small perturbation of that of the
Barenblatt solution. The pointwise convergence rates of the density, velocity
and the expanding rate of the physical vacuum boundary are also given. The
proof is based on space-time weighted energy estimates, elliptic estimates and
Hardy inequality in the Lagrangian coordinates. Our result is an extension of
that in Luo-Zeng [Comm. Pure Appl. Math. 69 (2016), no. 7, 1354-1396], where
the authors considered the physical vacuum free boundary problem of the
compressible Euler equations with constant-coefficient damping.
