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

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 \(\frac{\xi \gamma }{\chi \alpha }>M^{*}\).