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

Abstract

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source

$$\begin{aligned} \left\{ \begin{array}{llll} u_t = \Delta \left( u v^\alpha \right) +au-bu^l,\quad &{}x\in \Omega ,t>0,\\ v_t= \Delta v - vw,\quad &{}x\in \Omega ,t>0,\\ w_t = - \delta w + u,\quad &{}x\in \Omega ,t>0, \end{array} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) (\(n\ge 1\)). Here, the parameters \(a>0\), \(b>0\), \(\alpha \ge 1\), \(\delta >0\) and \(l \ge 2\). For all suitably regular initial data, if one of the following cases holds:

1. (i)

   \(l > 2\);

2. (ii)

   \(l =2, n\le 3\);

3. (iii)

   \(l = 2, n \ge 4,\) and b is sufficiently large, then the corresponding initial boundary value problem possesses a global classical solution.