In this paper, we are interested in the upper bound of the lifespan estimate for the compressible Euler system with time-dependent damping and small initial perturbations. We employ some techniques from the blow-up study of nonlinear wave equations. The novelty consists in the introduction of tools from the Orlicz spaces theory to handle the nonlinear term emerging from the pressure \(p \equiv p(\rho )\), which admits different asymptotic behavior for large and small values of \(\rho -1\), being \(\rho \) the density. Hence, we can establish, in dimensions \(n\in \{2,3\}\), unified upper bounds of the lifespan estimate depending only on the dimension n and on the damping strength and independent of the adiabatic index \(\gamma >1\). We conjecture our results to be optimal. The method employed here not only improves the known upper bounds of the lifespan for \(n\in \{2,3\}\), but has potential application in the study of related problems.

在本文中，我们感兴趣的是具有瞬态阻尼和小初始扰动的可压缩欧拉系统的寿命估计上限。我们采用了非线性波动方程的放大研究中的一些技术。新颖之处在于引入了 Orlicz 空间理论中的工具来处理从压力\(p \equiv p(\rho )\)中出现的非线性项，它允许较大和较小的\(\rho值) 的不同渐近行为-1\)，是\(\rho \)密度。因此，我们可以在维度\(n\in \{2,3\}\)中建立仅取决于维度n的寿命估计的统一上限以及阻尼强度，且与绝热指数\(\gamma >1\)无关。我们推测我们的结果是最佳的。这里采用的方法不仅提高了\(n\in \{2,3\}\)的已知寿命上限，而且在相关问题的研究中具有潜在的应用价值。