A lower bound for blow-up time in a fully parabolic Keller–Segel system

In this paper, an explicit lower bound for the blow-up time is obtained to a parabolic–parabolic Keller–Segel system, the blow-up conditions of which were established with an upper bound of blow-up time by Cieślak and Stinner [T. Cieślak, C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions, J. Differential Equations 252 (2012) 5832–5851].