Global Dynamics for a Class of Chemotaxis Systems with Density-Suppressed Motility and Nonlinear Indirect Signal Consumption

The paper is concerned with a chemotaxis model with nonlinear indirect signal consumption and density-suppressed motility

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (\varphi (v)u)+f(u),&x\in \Omega ,t>0,\\&v_t=\Delta v-vw^\beta ,&x\in \Omega ,t>0,\\&w_t=-\delta w+u,&x\in \Omega ,t>0, \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\) \((n\ge 1)\), where the parameters \(\delta \), \(\beta >0\), and \(\varphi (v)\) is a motility function. The purpose of this paper is to determine the size of the absorption exponent to ensure the existence of global bounded classical solutions to the problem. Specifically, we first showed that when \(f(u)=0\), the system has a global bounded classical solution if \(\beta \le 2\), \(n=1\), or \(\beta <\frac{4}{n}\), \(n\ge 2\), or suitably small initial data. Subsequently, when \(f(u)=ru-\mu u^\alpha \) with \(r\in \mathbb {R}\), \(\mu >0\), \(\alpha >1\), it was shown that the system admits a global bounded classical solution if \(\beta \le \alpha \), \(n=1\) or \(\beta <\max \bigl \{\alpha -1, \frac{2\alpha }{n}\bigr \}\), \(n\ge 2\), and that in the critical case \(\beta =\alpha -1\), \(n\ge 2\), we proved the existence of global bounded classical solutions provided that \(\mu \) is properly large. Moreover, we obtained the uniform convergence of bounded solutions to the system by constructing some suitable functionals.