On effects of the nonlinear signal production to the boundedness and finite-time blow-up in a flux-limited chemotaxis model

Abstract

We study herein the initial–boundary value problem for the flux-limited chemotaxis model with nonlinear signal production [Formula: see text] subject to no-flux boundary conditions in a ball [Formula: see text], where [Formula: see text], [Formula: see text]. For radially symmetric and positive initial data [Formula: see text], it is proved that the corresponding solution is globally bounded when [Formula: see text]. This result not only fills up a gap left in [M. Mizukami, T. Ono and T. Yokota, Extensibility criterion ruling out gradient blowup in a quasilinear degenerate chemotaxis system, J. Differ. Equ. 267(2019) 5115–5164; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259] when [Formula: see text], but also extend the boundedness result from the special case [Formula: see text] to the general case [Formula: see text]. Moreover, under the condition [Formula: see text] it is shown that if [Formula: see text] is sufficiently large, then there exists initial data [Formula: see text] such that the corresponding solution blows up at finite time [Formula: see text] in the sense that [Formula: see text] This blow-up result generalizes the works for the linear signal production case in (N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Am. Math. Soc. Ser. B 4 (2017) 31–67; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259) to the nonlinear case [Formula: see text].