Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism

Abstract

In this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism$ \begin{equation*} \left\{ \begin{array}{ll} v_{1t} = \nabla \cdot\big(\psi(v_{1})\nabla v_{1}-\chi \phi(v_{1})\nabla v_{2}\big)+\lambda_{1}v_{1}-\lambda_{2}v_{1}^{\beta},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{2t} = \Delta v_{2}-w^{\theta}v_{2}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+v_{1}^{\alpha}, \ &\ \ x\in \Omega, \ t>0 ,\\[2.5mm] \end{array} \right. \end{equation*} $in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \chi, \; \lambda_{1}, \; \lambda_{2}, \; \theta > 0, \; 0 < \alpha\leq\frac{1}{\theta}, \; \beta\geq 2, \; $ $ \psi $, and $ \phi $ are nonlinear functions that satisfy $ \psi(s)\geq a_{0}(s+1)^{r_{1}} $ and $ 0\leq\phi(s)\leq b_{0}s(s+1)^{r_{2}} $ for all $ s\geq 0 $ with $ a_{0}, b_{0} > 0 $ and $ r_{1}, r_{2}\in \mathbb{R}. $ It has been proven that if $ r_{1} > 2r_{2}+1, $ then the problem admits a global and bounded classical solution for some appropriate nonnegative initial data.

Keywords

• chemotaxis system
• global boundedness
• nonlinear indirect signal