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The large time behavior of entropy solution to the compressible Eulerequations 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 methodand using the intensive entropy analysis, it is proved that the $L^\infty$entropy solution of compressible Euler equations with finite initial massconverges strongly in the natural $L^1$ topology to a fundamental solution ofporous media equation (PME) with time-dependent diffusion, called bygeneralized Barenblatt solution. It is interesting that the $L^1$ decay rate isgetting faster and faster as $\lambda$ increases in $(0,\frac{\gamma}{\gamma+2}]$, while is getting slower and slower in $[\frac{\gamma}{\gamma+2}, 1)$.

$L^1$-convergence to generalized Barenblatt solution for compressible Euler equations with time-dependent damping