Global boundedness in a chemotaxis system with signal-dependent motility and indirect signal consumption and logistic source

This paper deals with the chemotaxis system with signal-dependent motility and indirect signal consumption and logistic source$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta ( u\gamma(v))+au-bu^l, \quad &x\in \Omega, t>0, \\ v_t = \Delta v - vw, \quad &x\in \Omega, t>0, \\ w_t = - \delta w + u, \quad &x\in \Omega, t>0, \end{array} \right. \end{eqnarray*} $under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega\subset \mathbb{R}^n $ ($ n\geq1 $). Here the parameters $ \delta >0 $, $ a>0 $, $ b>0 $ and $ l>1 $, and the motility function $ \gamma \in {C^3}\left( {\left[ {0, + \infty } \right)} \right) $ is positive on $ [0, \infty) $. For all suitably regular initial data, if one of the following cases holds:(ⅰ) $ n \le 3, l > 1 $;(ⅱ) $ n \ge 4, l > 2 $;(ⅲ) $ n \ge 4, l = 2 $ and $ b $ is sufficiently large,then the corresponding initial boundary value problem possesses global bounded classical solutions. Our goal extends the global solution result by Li et al. [19] with sufficiently large values of $ l > \max \{ 1, \frac{n}{2}\} $.