Boundedness criteria for the quasilinear attraction-repulsion chemotaxis system with nonlinear signal production and logistic source

Abstract

This paper deals with the following quasilinear attraction-repulsion chemotaxis system $ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \nabla\cdot((u+1)^{m}\nabla u-\chi u(u+1)^{\theta-1}\nabla v+\xi u(u+1)^{l-1}\nabla w)+au-bu^{\kappa}, \ &\ \ x\in \Omega, \ t>0, \\ 0 = \Delta v+\alpha u^{\gamma_{1}}-\beta v, \ &\ \ x\in \Omega, \ t>0, \\ 0 = \Delta w+\gamma u^{\gamma_{2}}-\delta w, \ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $ with homogeneous Neumann boundary conditions in a bounded, smooth domain $ \Omega\subset\mathbb{R}^{n} (n\geq 1), $ where $ m, \theta, l \in \mathbb{R} $ and $ \chi, \xi, a, b, \alpha, \beta, \gamma, \delta, \gamma_{1}, \gamma_{2} > 0, \kappa > 1. $ It is proved that if the nonlinear exponents of the system satisfy $ \theta+\gamma_{1} < \max\left\{l+\gamma_{2}, \kappa, m+\frac{2}{n}+1\right\}, $ then the system has globally bounded classical solutions. Furthermore, assume that $ \theta+\gamma_{1} = \max\{l+\gamma_{2}, \kappa\}\geq m+\frac{2}{n}+1, $ if one of the following conditions holds: $ \begin{align*} \mbox{(a)}\ \mbox{when}\ \theta+\gamma_{1} = l+\gamma_{2} = \kappa, & \ \mbox{if}\ \theta\geq l\geq 1 \ \mbox{and} \ \frac{[(\kappa-1-m)n-2](2\alpha \chi-\gamma\xi)}{2(l-1)+(\kappa-1-m)n}<b \\ \ &\mbox{or if}\ \ l \geq \theta\geq 1\ \mbox{and} \ \frac{2\alpha \chi[(\kappa-1-m)n-2]}{2(\theta-1)+(\kappa-1-m)n}<b; \\ \ \mbox{(b)}\ \mbox{when}\ \theta+\gamma_{1} = l+\gamma_{2}>\kappa, & \ \mbox{if}\ \theta\geq l\geq 1 \ \mbox{and} \ 2\alpha \chi\leq \gamma\xi; \\ \ \mbox{(c)}\ \mbox{when}\ \theta+\gamma_{1} = \kappa>l+\gamma_{2}, & \ \mbox{if}\ \theta\geq 1 \ \mbox{and} \ \frac{2\alpha \chi[(\kappa-1-m)n-2]}{2(\theta-1)+(\kappa-1-m)n}<b, \end{align*} $ then the classical solutions of the system would be globally bounded. The global boundedness criteria generalize the results established by previous researchers.

Keywords

• boundedness criteria
• attraction-repulsion system
• nonlinear signal production
• logistic source