Logistic damping effect in a chemotaxis system with density-suppressed motility and indirect signal consumption

In this paper, we study the following chemotaxis model $\left(\begin{array}{cc}{u}_t=\Delta \left(\phi \left(v\right)u\right)+ru-\mu u^l,& x\in \Omega ,t>0,\\ v_t=\Delta v-vw,& x\in \Omega ,t>0,\\ w_t=-\delta w+u,& x\in \Omega ,t>0,\end{array}\right)$under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset R^n(n\ge 1)$ with smooth boundary, where the parameters $\delta$, $\mu >0$, $r\in R$ and $l>1$. The positive motility function satisfies $\phi \in C^3\left([0,\infty )\right)$, and the purpose of this paper is to weaken the restriction on $l$ which ensures the existence of global bounded solutions Li et al. (2022). It was shown that when $n\le 3$, there exists a global bounded classical solution for all $l>1$. When $n\ge 4$, then we concluded that the system admits a global bounded classical solution for all $l>2$, and that the sufficiently large $\mu$ can ensure the existence of global bounded solutions if $l=2$. Moreover, we also studied the large time behavior of solutions.