Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in ²

This paper concerns the long time existence to the smooth solutions of the compressible Euler system with critical time dependent damping in ². We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. For this end, the vector fields Z (defined below) are used instead of the usual one Z, to get better decay for the linear error terms. This idea may also apply to the long time behavior study of nonlinear wave equations with time-dependent damping.