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 \(\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (u\nabla v)+\mu u \left( 1-\int _{\Omega }u^{\alpha }\text {d}x\right) ,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] 0=\Delta v-m(t)+u,\ m(t)=\frac{1}{|\Omega |}\int _{\Omega } u(x,t)\text {d}x,\ {} &{}\ \ x\in \Omega , \ t>0,\\[2.5mm] u(x,0)=u_{0}(x), \ {} &{}\ \ x\in \Omega , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\nabla }·(\phi (u)\mathrm{\nabla }u)-\mathrm{\nabla }·(u\mathrm{\nabla }v)+\mu u\left(1,-,{\int }_{\mathrm{\Omega }},u^{\alpha },\text{d},x\right),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ [2.5mm]0=\mathrm{\Delta }v-m(t)+u,m(t)=\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}u(x,t)\text{d}x,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ [2.5mm]u(x,0)=u_0(x),\hfill & x\in \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega =B_{R}(0)\subset {\mathbb {R}}^n (n\ge 3)\) $\mathrm{\Omega }=B_R(0)\subset {\mathbb{R}}^n(n\ge 3)$ with \(R>0,\) $R>0,$ the parameters \(\mu , \alpha \) $\mu ,\alpha$ are positive constants and diffusion function \( \varphi (u)\le C_{0}(1+u)^{-m}\) $\phi (u)\le C_0{(1+u)}^{-m}$ for all \(u\ge 0\) $u\ge 0$ with \(C_{0}>0\) $C_0>0$ and \(m> -1.\) $m>-1.$ It has been shown that if \(\begin{aligned} 0<\alpha <\min \left\{ 2,\frac{n}{2},\frac{n(m+1)}{2}\right\} , \end{aligned}\) $$\begin{array}{c}\hfill 0<\alpha <min\left\{2,,,\frac{n}{2},,,\frac{n(m+1)}{2}\right\},\end{array}$$ then there exist suitable initial data \(u_{0}\) $u_0$ 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.