In this paper, we consider the following attraction repulsion chemotaxis model with nonlinear signal term: \begin{ align*} &u_{ t}=\nabla \cdot(\nabla u-\xi_{ 1} u \nabla v +\xi_{ 2} u \nabla w), \quad &0=\Delta v -\lambda_{ 1}v +f_{ 1}(u), \quad &0=\Delta w -\lambda_{ 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 \geqslant 1$, or $l=m<\frac{ 2}{ N}$.
Categories: math.AP