Global classical solutions to an indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source \(\begin{aligned} \left\{ \begin{array}{llll} u_t = \Delta \left( u v^\alpha \right) +au-bu^l,\quad &{}x\in \Omega ,t>0,\\ v_t= \Delta v - vw,\quad &{}x\in \Omega ,t>0,\\ w_t = - \delta w + u,\quad &{}x\in \Omega ,t>0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\Delta }\left(u,v^{\alpha }\right)+au-b{u}^l,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=-\delta w+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( \(n\ge 1\) $n\ge 1$ ). Here, the parameters \(a>0\) $a>0$ , \(b>0\) $b>0$ , \(\alpha \ge 1\) $\alpha \ge 1$ , \(\delta >0\) $\delta >0$ and \(l \ge 2\) $l\ge 2$ . For all suitably regular initial data, if one of the following cases holds: (i)

\(l > 2\) $l>2$ ;