Blow-up Analysis to a Quasilinear Chemotaxis System with Nonlocal Logistic Effect

In this paper, we consider the following quasilinear chemotaxis system involving nonlocal effect

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (u\nabla v)+\mu u \left( 1-\int _{\Omega }u^{\alpha }\text {d}x\right) ,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] u(x,0)=u_{0}(x), \ {} &{}\ \ x\in \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)\) with \(R>0,\) the parameters \(\mu , \alpha \) are positive constants and diffusion function \( \varphi (u)\le C_{0}(1+u)^{-m}\) for all \(u\ge 0\) with \(C_{0}>0\) and \(m> -1.\) It has been shown that if

$$\begin{aligned} 0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\} , \end{aligned}$$

then there exist suitable initial data \(u_{0}\) such that the corresponding radially symmetric solution blows up in finite time. In this work, we extend the blow-up result established by previous researchers.