An Attraction-Repulsion Chemotaxis System: The Roles of Nonlinear Diffusion and Productions

Abstract

This article considers the no-flux attraction-repulsion chemotaxis model

$$ \left \{ \textstyle\begin{array}{l} \begin{aligned} &u_{t} = \nabla \cdot \big((u+1)^{m_{1}-1}\nabla u-\chi u(u+1)^{m_{2}-2} \nabla v+\xi u(u+1)^{m_{3}-2}\nabla w\big),& x\in \Omega ,\ t>0&, \\ & 0=\Delta v+f(u)-\beta v, & x\in \Omega ,\ t>0&, \\ & 0=\Delta w+g(u)-\delta w, & x\in \Omega ,\ t>0& \end{aligned} \end{array}\displaystyle \right . $$

defined in a smooth and bounded domain $\Omega \subset \mathbb{R}^{n}$ ($n\ge 2$) with $m_{1},m_{2},m_{3}\in \mathbb{R}$, $\chi ,\xi ,\beta ,\delta >0$. The functions $f(u)$, $g(u)$ extend the prototypes $f(u)=\alpha u^{s}$ and $g(u)=\gamma u^{r}$ with $\alpha ,\gamma >0$ and suitable $s,r>0$ for all $u\ge 0$. Our main result exhibits that there exists $M^{*}>0$ such that for all properly regular initial data, the studied model admits a unique classical solution which remains bounded if $m_{2}+s< m_{3}+r$ or $m_{2}+s=m_{3}+r$ and $\frac{\xi \gamma }{\chi \alpha }>M^{*}$.