# American Institute of Mathematical Sciences

May  2021, 26(5): 2537-2559. doi: 10.3934/dcdsb.2020194

## A blow-up result for the chemotaxis system with nonlinear signal production and logistic source

 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author

Received  October 2019 Revised  March 2020 Published  May 2021 Early access  June 2020

In this paper we consider the following chemotaxis-growth system with nonlinear signal production and logistic source
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad &x\in \Omega, t>0, \\ 0 = \Delta v-\mu (t)+f(u), \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}f(u(\cdot, t)), \quad &x\in \Omega, t>0, \ \end{array} \right. \end{eqnarray*}$
with homogeneous Neumann boundary conditions in the ball
 $\Omega = B_R(0)\subset \mathbb{R}^n$
for
 $n\geq 1$
and
 $R>0$
, where
 $\chi, \lambda, \mu>0$
,
 $\alpha>1$
, and
 $f$
is an appropriate regular function satisfying
 $f(u)\geq ku^{\kappa}$
for all
 $u\geq1, \kappa>0$
with some constants
 $k>0$
. If the number
 $\kappa$
and
 $\alpha$
satisfy
 $\begin{equation*} \kappa+1>\alpha\left(\frac{2}{n}+1\right), \end{equation*}$
then there exists appropriate initial data such that the corresponding solution
 $(u, v)$
of the system blow up in finite time. This result extends the blow-up result of the chemotaxis model without logistic cell kinetics in [45]. Apparently, for the case
 $\kappa = 1$
, this provides a rigorous detection for blow-up of solution in spaces-dimensions three and four.
Citation: Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
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