Analysis of global dynamics in an attraction-repulsion model with nonlinear indirect signal and logistic source

The following chemotaxis system has been considered:

$ \begin{equation*} \left\{ \begin{array}{ll} v_{t} = \Delta v-\xi \nabla\cdot(v \nabla w_{1})+\chi \nabla\cdot(v \nabla w_{2})+\lambda v-\mu v^{\kappa},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] w_{1t} = \Delta w_{1}-w_{1}+w^{\kappa_{1}}, \ 0 = \Delta w-w+v^{\kappa_{2}}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w_{2}-w_{2}+v^{\kappa_{3}}, \ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $

under the boundary conditions of $ \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w_{1}}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = \frac{\partial{w_{2}}}{\partial{\nu}} $ on $ \partial \Omega, $ where $ \Omega $ was a bounded smooth domain of $ \mathbb{R}^{n}(n\geq 1), \; \nu $ was the normal vector of $ \partial\Omega, $ and the parameters were $ \lambda, \mu, \xi, \chi, \kappa_{1}, \; \kappa_{2}, \kappa_{3} > 0, $ and $ \kappa > 1. $ In this paper, we showed that if either $ \kappa_{1}\kappa_{2} < \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ or $ \kappa_{1}\kappa_{2} = \max\{\frac{2}{n}, \kappa_{3}, \kappa-1\} $ with the coefficients and initial data satisfying appropriate conditions, then the system possessed a global classical solution. Furthermore, we also have studied the convergence of solutions to a special case of the above system with $ \kappa = \delta+1, \kappa_{1} = 1, \kappa_{2} = \kappa_{3} = \delta $ for $ \delta > 0. $ It has been proven that if $ \mu > 0 $ is large enough, then the corresponding classical solutions exponentially converged to $ ((\frac{\lambda}{\mu})^{\frac{1}{\delta}}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}, \frac{\lambda}{\mu}), $ where the convergence rate could be formally expressed by the parameters of the system.