Singularity formation for compressible Euler equations with time-dependent damping

In this paper, we consider the compressible Euler equations with time-dependent damping \begin{document}$ \frac{{\alpha}}{(1+t)^\lambda}u $\end{document} in one space dimension. By constructing "decoupled" Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient grows with a algebraic rate. We study the case \begin{document}$ \lambda\neq1 $\end{document} and \begin{document}$ \lambda = 1 $\end{document} respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for \begin{document}$ 1 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.