Boundedness and finite-time blow-up in a quasilinear chemotaxis system with space dependent logistic source and nonlinear production

Abstract

This paper is concerned with the following quasilinear chemotaxis system with space dependent logistic source and nonlinear production ($\star$)$\left(\begin{array}{cccc}& u_t=\nabla \cdot \left(\phi \left(u\right)\nabla u\right)-\chi \nabla \cdot \left(\psi \left(u\right)\nabla v\right)+\lambda \left(\right|x\left|\right)u-\kappa \left(\right|x\left|\right)u^{\theta },& x& \in B_R,t>0,\\ & 0=\Delta v-\mu \left(t\right)+f\left(u\right),\mu \left(t\right)=\frac{1}{\left|B_R\right|}{\int }_{B_R}f\left(u(x,t)\right)dx,& x& \in B_R,t>0,\end{array}\right)$with homogeneous Neumann boundary conditions, where $B_R=B_R\left(0\right)\subset R^n$ with $n⩾1$, $R>0$ and $\theta >1$. Here $\lambda$ and $\kappa$ are continuous positive functions, $-{\lambda }^{\prime },{\kappa }^{\prime }>0$ and $\kappa$ has a positive lower bound, the nonlinear diffusivity $\phi \left(u\right)$, chemosensitivity $\psi \left(u\right)$ and signal production $f\left(u\right)$ are supposed to extend the prototypes $\phi \left(u\right)={(u+1)}^{-p},\psi \left(u\right)=u{(u+1)}^{q-1},\text{and}f\left(u\right)=u^l,$with $p\in R$ and $q,l>0$. Under some suitable assumptions on $\phi \left(u\right)$, $\psi \left(u\right)$ and $f\left(u\right)$, we show that there exist the global boundedness and finite-time blow-up of solutions for the problem ($\star$).