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

This paper concerns the long time existence of the smooth solutions of the compressible Euler system with critical time-dependent damping in R 2 . We establish the sharp lifespan estimate from below, with respect to the small parameter of the initial perturbation. In order to be adapted to the time-dependent damping term, the vector field S + 1 is used instead of the standard scaling vector S , which results in better decay estimates for the linear error terms. This idea may also be applied to the long time behavior study of nonlinear wave equations with time-dependent damping.