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

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.