Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source

Abstract

This paper deals with the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source,

$$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v^\gamma }\nabla v)+\lambda u-\mu u^k,\quad &{}x\in \Omega ,\quad t>0,\\ 0=\Delta v-v+u,\quad &{}x\in \Omega ,\quad t>0\\ \end{array} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions and suitable initial conditions where $\Omega =B_R(0)\subset \mathbb {R}^n$ ($n\ge 3$, $R>0$), $\chi >0$, $\gamma >0$, $\lambda \in \mathbb {R}$, $\mu >0$, $k>1$. In the following two cases:

$$\begin{aligned} \begin{aligned} \bullet ~\left\{ \begin{array}{llll} \frac{7-\sqrt{37}}{2}<\gamma<\frac{1}{2}\quad &{}\text {if}\,\ n\in \{3,4\},\\ \frac{33-\sqrt{969}}{4}<\gamma<\frac{1}{2}\quad &{}\text {if}\,\ n=6, \end{array} \right. \text {and}~~~ \frac{1}{2\gamma }\le k< \left\{ \begin{array}{llll} 1+\frac{1-\gamma }{6}\quad &{}\text {if}\,\ n\in \{3,4\},\\ 1+\frac{3-2\gamma }{30}\quad &{}\text {if}\,\ n=6; \end{array} \right. \end{aligned} \end{aligned}$$

$\bullet$ $\frac{1}{2}\le \gamma <1$ and $1<k<1+\frac{1-\gamma }{6}$ with $n\in \{3,4\}$, it is proved that the corresponding solution blows up in finite time, which extends the blow-up result of Winkler [44] to the case with singular sensitivity.