  The large time behavior of entropy solution to the compressible Euler
equations for polytropic gas (the pressure $p(\rho)=\kappa\rho^{\gamma},
\gamma>1$) with time dependent damping like $-\frac{1}{(1+t)^\lambda}\rho u$
($0<\lambda<1$) is investigated. By introducing an elaborate iterative method
and using the intensive entropy analysis, it is proved that the $L^\infty$
entropy solution of compressible Euler equations with finite initial mass
converges strongly in the natural $L^1$ topology to a fundamental solution of
porous media equation (PME) with time-dependent diffusion, called by
generalized Barenblatt solution. It is interesting that the $L^1$ decay rate is
getting faster and faster as $\lambda$ increases in $(0,
\frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[
\frac{\gamma}{\gamma+2}, 1)$.
