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

This article considers the no-flux attraction-repulsion chemotaxis model $\{\begin{array}{l}\begin{array}{rlrl}& u_t=\mathrm{\nabla }\cdot ({(u+1)}^{m_1-1}\mathrm{\nabla }u-\chi u{(u+1)}^{m_2-2}\mathrm{\nabla }v+\xi u{(u+1)}^{m_3-2}\mathrm{\nabla }w),& x\in \mathrm{\Omega },t>0& ,\\ & 0=\mathrm{\Delta }v+f(u)-\beta v,& x\in \mathrm{\Omega },t>0& ,\\ & 0=\mathrm{\Delta }w+g(u)-\delta w,& x\in \mathrm{\Omega },t>0& \end{array}\end{array}$ \( \left \{ \textstyle\begin{array}{l} \begin{aligned} &u_{t} = \nabla \cdot \big((u+1)^{m_{1}-1}\nabla u-\chi u(u+1)^{m_{2}-2} \nabla v+\xi u(u+1)^{m_{3}-2}\nabla w\big),& x\in \Omega ,\ t>0&, \\ & 0=\Delta v+f(u)-\beta v, & x\in \Omega ,\ t>0&, \\ & 0=\Delta w+g(u)-\delta w, & x\in \Omega ,\ t>0& \end{aligned} \end{array}\displaystyle \right . \) defined in a smooth and bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $\Omega \subset \mathbb{R}^{n}$ ( $n\ge 2$ $n\ge 2$ ) with $m_1,m_2,m_3\in \mathbb{R}$ $m_{1},m_{2},m_{3}\in \mathbb{R}$ , $\chi ,\xi ,\beta ,\delta >0$ $\chi ,\xi ,\beta ,\delta >0$ . The functions $f(u)$ $f(u)$ , $g(u)$ $g(u)$ extend the prototypes $f(u)=\alpha u^s$ $f(u)=\alpha u^{s}$ and $g(u)=\gamma u^r$ $g(u)=\gamma u^{r}$ with $\alpha ,\gamma >0$ $\alpha ,\gamma >0$ and suitable $s,r>0$ $s,r>0$ for all $u\ge 0$ $u\ge 0$ . Our main result exhibits that there exists $M^{\ast }>0$ $M^{*}>0$ such that for all properly regular initial data, the studied model admits a unique classical solution which remains bounded if $m_2+s<m_3+r$ $m_{2}+s< m_{3}+r$ or $m_2+s=m_3+r$ $m_{2}+s=m_{3}+r$ and $\frac{\xi \gamma }{\chi \alpha }>M^{\ast }$ $\frac{\xi \gamma }{\chi \alpha }>M^{*}$ .