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In this paper, we consider some blow-up problems for the 1D Euler equationwith time and space dependent damping. We investigate sufficient conditions oninitial data and the rate of spatial or time-like decay of the coefficient ofdamping for the occurrence of the finite time blow-up. In particular, oursufficient conditions ensure that the derivative blow-up occurs in finite timewith the solution itself and the pressure bounded. Our method is based onsimple estimates with Riemann invariants. Furthermore, we give sharp lower andupper estimates of the lifespan of solutions, when initial data are smallperturbations of constant states.

Singularity formation for the 1D compressible Euler equation with variable damping coefficient