Global boundedness for a chemotaxis system involving nonlinear indirect consumption mechanism.

We study the following chemotaxis system$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi\nabla \cdot (u\nabla v)+\eta(u-u^{m}),\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] v_{t} = \Delta v-u^{\theta}vw, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w-w+u^{\alpha}, \ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $in a smooth and bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 2) $ under homogeneous Neumann boundary conditions, where $ \chi, \eta, \alpha,\theta>0 $ and $ m>1. $ It has been shown that if $ l_{0} = \max\{\alpha,\theta\} $ and $ m $ satisfy$ \begin{equation*} \left\{ \begin{array}{ll} m>l_{0}(n+2),\ &\ \ \mbox{if} \ l_{0}>\frac{1}{n+2},\\[2.5mm] m>1, \ &\ \ \mbox{if} \ 0<l_{0}\leq\frac{1}{n+2}, \end{array} \right. \end{equation*} $then the corresponding initial-boundary value problem admits a classical solution which is bounded in $ \Omega \times (0,\infty). $ [ABSTRACT FROM AUTHOR]