Nonlinear generation index of bounded solutions in a chemotaxis system with indirect signal, density-suppressed motility and generalized logistic source

In this paper, we study the following chemotaxis-growth system with density-suppressed motility$\{\begin{array}{ccc}& u_t=\Delta \left(u\gamma \right(v\left)\right)+ru-\mu u^{\alpha },& x\in \Omega ,t>0,\\ & v_t=\Delta v-v+w^{\beta },& x\in \Omega ,t>0,\\ & w_t=-\delta w+u,& x\in \Omega ,t>0,\end{array}$ under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset R^n(n\ge 2)$ with smooth boundary, where $r\in R$, μ, δ, $\beta >0$, $\alpha >1$, and the positive motility function $\gamma \in C^3\left(\right[0,\infty \left)\right)$ satisfies ${\gamma }^{\prime }\left(s\right)\le 0$ for all $s\ge 0$. The main purpose of this paper is to establish the global boundedness and stabilization of classical solutions for such kind of chemotaxis system. More precisely, we showed that if $\beta <\frac{2\alpha }{n}$, the system possesses a globally bounded classical solution, and further, if γ also fulfills the additional assumption that $\frac{{\gamma }^{\prime }}{\gamma }$ is bounded on $[0,\infty )$, then for $\alpha >\frac{n}{n-2}$, the above restriction can be optimized as $\beta <\frac{(n+2)\alpha -n}{2n}$. Furthermore, we obtained that the solution will converge to the equilibrium $({\left(\frac{r_{+}}{\mu }\right)}^{\frac{1}{\alpha -1}},\frac{1}{{\delta }^{\beta }}{\left(\frac{r_{+}}{\mu }\right)}^{\frac{\beta }{\alpha -1}},\frac{1}{\delta }{\left(\frac{r_{+}}{\mu }\right)}^{\frac{1}{\alpha -1}})$ with an exponential decay rate in ${\left(L^{\infty }\right(\Omega \left)\right)}^3$, where $r_{+}:=\max \{r,0\}$.