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Singularity formation for compressible Euler equations with time-dependent damping

In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^λ}u in one space dimension. By constructing &#39;decoupled&#39; 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 goes to infinity with a algebraic growth rate. We study the case λ\neq1 and λ=1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1<γ<3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.