Vacuum and singularity formation for compressible Euler equations with time-dependent damping

In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $ 1<\gamma{\leq} 3 $, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $ \lambda $, which was open in [19] for some cases. Moreover, the singularity formation of the compressible Euler equations when $ \gamma = 3 $ is investigated, too.