Boundedness in an attraction-repulsion chemotaxis system with indirect repulsion-signal production

This paper studies the following attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanism$\{\begin{array}{ccc}& u_t=\Delta u-\xi \nabla \cdot \left(u\nabla v\right)+\chi \nabla \cdot \left(u\nabla z\right)+f\left(u\right),& x\in \Omega ,t>0,\\ & 0=\Delta v-v+u^{{\gamma }_1},& x\in \Omega ,t>0,\\ & z_t=\Delta z-z+w^{{\gamma }_2},& x\in \Omega ,t>0,\\ & 0=\Delta w-w+u^{{\gamma }_3},& x\in \Omega ,t>0,\end{array}$ under homogeneous Neumann boundary conditions, where $\Omega \subset R^n(n\ge 1)$ is a smoothly bounded domain and $\xi ,\chi ,{\gamma }_1,{\gamma }_2,{\gamma }_3>0$.

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  When $f\equiv 0$, it is shown that the solution of the above system is global and uniformly bounded if $\max \{{\gamma }_1,{\gamma }_2{\gamma }_3\}<\frac{2}{n}$. Moreover, if $\max \{{\gamma }_1,{\gamma }_2{\gamma }_3\}=\frac{2}{n}$, the boundedness of solution can be derived provided that the initial mass ${\int }_{\Omega }{u}_0\left(x\right)dx$ is small.
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  When $f\left(u\right)\le u(a-b{u}^k)$ with $a,b,k>0$, it is proved that if one of the following conditions holds: (i) $k>\max \{{\gamma }_1,{\gamma }_2{\gamma }_3\}$, (ii) $k=\max \{{\gamma }_1,{\gamma }_2{\gamma }_3\}$, (iii) $\max \{{\gamma }_1,{\gamma }_2{\gamma }_3\}<\frac{2}{n}$, then the solution is globally bounded in time provided that b is large enough.