Estimate To Lower Bound Of Blow-up Time In A Parabolic-parabolic Keller-Segel Model

The so-called Keller-Segel system is a famous model in mathematical biology, describing the chemotaxis phenomena in evolutions of populations. This system belongs to parabolic-elleptic or parabolic-parabolic equations with cross-diffusion mechanism. This paper deals with the asymptotic behavior of solutions to a parabolic-parabolic Keller-Segel system. Based on the known results on the blow-up conditions and the upper bound estimate of blow-up time [T. Ciesak, C. Stinner, Finitetime blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations252(2012)5832-5851], we establish lower bound estimate of blow-up time. This contributes to the studies of Keller-Segel system.