Global bounded solution of the chemotaxis attraction repulsion Cauchy problem with the nonlinear signal production in ${\mathbb{R}}^N$

In this paper, we consider the following attraction repulsion chemotaxis model with nonlinear signal term: \begin{align*} &u_{t}=\nabla \cdot(\nabla u-ξ_{1} u \nabla v +ξ_{2} u \nabla w), \quad &0=Δv -λ_{1}v +f_{1}(u), \quad &0=Δw -λ_{2}w +f_{2}(u), \quad x \in $\mathbb{R}$^{N}, t>0, \end{align*} where ${\xi }_1,{\xi }_2,{\lambda }_1,{\lambda }_2$ are for some positive constants, and \begin{equation*} f_{1} \in C^{1}([0,\infty)) ; $\text{satisfying}$ ; 0 \leqslant f_{1}(s) \leqslant c_{1}s^{l}, ; \forall s \geqslant 0 \ $\text{and}$ \ l> 0, \end{equation*} \begin{equation*} f_{2} \in C^{1}([0,\infty)); $\text{satisfying}$ ; 0 \leqslant f_{2}(s) \leqslant c_{2}s^{m}, ; \forall s \geqslant 0 \ $\text{and}$ \ m> 0. \end{equation*} We will show that this problem has a unique global bounded solution when $l>\frac{2}{N},l<m\text{ with }m⩾1$, or $l=m<\frac{2}{N}$.