This talk deals with the Cauchy problem for the  compressible Euler equations with time-dependent damping in one-dimensional space, where the time-vanishing damping makes some fantastic variety of the  dynamic system. For slow damping, the solutions are proved to exist globally in time, when the derivatives of the initial data are small, but the initial data themselves can be  arbitrarily large. When   the initial Riemann invariants are monotonic and their derivatives with absolute value  are large at least at one point, then the solutions are still bounded, but their derivatives will blow up at finite time. For fast damping, the derivatives of solutions will  blow up even for all initial data, including the  interesting case of blow-up solutions with small initial data. Here the initial Riemann invariants are monotonic. In order to prove the global existence of solutions with large initial data, we introduce a new energy functional related the Riemann invariants, which crucially enables us to build up the maximum principle for the corresponding Riemann invariants, and the uniform boundedness for the local solutions. Finally,  numerical simulations in different cases are carried out, which further confirm our theoretical results.