Global boundedness in a Keller-Segel system with nonlinear indirect signal consumption mechanism

In this paper, we study a quasilinear chemotaxis model with a nonlinear indirect consumption mechanism

$$\{\begin{array}{ll}{v}_{1t}=\mathrm{\nabla }\cdot (\psi (v_1)\mathrm{\nabla }{v}_1-\chi \varphi (v_1)\mathrm{\nabla }{v}_2)+{\lambda }_1{v}_1-{\lambda }_2{v}_1^{\beta },& x\in \mathrm{\Omega },t>0,\\ v_{2t}=\mathrm{\Delta }{v}_2-w^{\theta }{v}_2,& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }w-w+v_1^{\alpha },& x\in \mathrm{\Omega },t>0,\end{array}$$

in a smooth and bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ with homogeneous Neumann boundary conditions, where $\chi ,{\lambda }_1,{\lambda }_2,\theta >0,0<\alpha \le \frac{1}{\theta },\beta \ge 2,$ $\psi$, and $\varphi$ are nonlinear functions that satisfy $\psi (s)\ge a_0(s+1{)}^{r_1}$ and $0\le \varphi (s)\le b_{0}s(s+1{)}^{r_2}$ for all $s\ge 0$ with $a_0,b_0>0$ and $r_1,r_2\in \mathbb{R}.$ It has been proven that if $r_1>2r_2+1,$ then the problem admits a global and bounded classical solution for some appropriate nonnegative initial data.