Global Boundedness in a Chemotaxis System Involving Signal-Dependent Motility and Indirect Signal Consumption

In this paper, we consider the following consumption chemotaxis system

$$ \textstyle\begin{cases} u_{t}=\Delta \left (u\phi (v)\right )+au-bu^{\gamma }, &(x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta {v}-uvw, & (x,t)\in \Omega \times (0,\infty ), \\ w_{t}=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{cases} $$

under the smooth bounded domain \(\Omega \subset \mathbb{R}^{n}\,\,(n\ge 2)\) with homogeneous Neumann boundary conditions, where the parameters \(a>0\), \(b>0\), \(\gamma \ge 2\) and \(\delta >0\). It has been shown that for any sufficiently regular initial data, the associated initial-boundary value problem has a global classical solutions.