Abstract: In this paper, we are concerned with the asymptotic behavior of $L^{\infty}$ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping $-\frac{m}{(1+t)^{\lambda}}$. As $\lambda \in (0,\frac17]$, we prove that the $L^{\infty}$ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in $L^2(\mathbb R)$. As $\lambda \in (\frac17,1)$, we prove that the $L^{\infty}$ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in $L^2(\mathbb R)$, which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.

Key words: L²-convergence, compressible Euler Equations, time asymptotic expansion, time-dependent damping, relative entropy inequality