Global existence and uniform boundedness to a bi-attraction chemotaxis system with nonlinear indirect signal mechanisms

In this paper, we study the following quasilinear chemotaxis system $ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \Delta u-\chi \nabla \cdot (\varphi (u)\nabla v)-\xi \nabla \cdot (\psi(u)\nabla w)+f(u), \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta v-v+v_{1}^{\gamma_{1}}, \ 0 = \Delta v_{1}-v_{1}+u^{\gamma_{2}}, \ &\ \ x\in \Omega, \ t>0, \ \\ 0 = \Delta w-w+w_{1}^{\gamma_{3}}, \ 0 = \Delta w_{1}-w_{1}+u^{\gamma_{4}}, \ &\ \ x\in \Omega, \ t>0, \end{array} \right. \end{equation*} $ in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{n}(n\geq 1) $ with homogeneous Neumann boundary conditions, where $ \varphi(\varrho)\leq\varrho(\varrho+1)^{\theta-1}, $ $ \psi(\varrho)\leq\varrho(\varrho+1)^{l-1} $ and $ f(\varrho)\leq a \varrho-b\varrho^{s} $ for all $ \varrho\geq0, $ and the parameters satisfy $ a, b, \chi, \xi, \gamma_{2}, \gamma_{4} > 0, $ $ s > 1, $ $ \gamma_{1}, \gamma_{3}\geq1 $ and $ \theta, l\in \mathbb{R}. $ It has been proven that if $ s \geq\max\{ \gamma_{1}\gamma_{2}+\theta, \gamma_{3}\gamma_{4}+l\}, $ then the system has a nonnegative classical solution that is globally bounded. The boundedness condition obtained in this paper relies only on the power exponents of the system, which is independent of the coefficients of the system and space dimension $ n. $ In this work, we generalize the results established by previous researchers.