Global dynamics to a quasilinear chemotaxis system under some critical parameter conditions

In this manuscript, the following chemotaxis system has been considered:

$ \begin{equation*} \left\{ \begin{array}{ll} v_{t} = \nabla\cdot(\phi(v)\nabla v-\varphi(v)\nabla w_{1}+\psi(v)\nabla w_{2})+av-bv^{\kappa},\ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w_{1}+\alpha v^{\gamma_{1}}-\beta w_{1}, \ &\ \ x\in \Omega, \ t>0,\\[2.5mm] 0 = \Delta w_{2}+\gamma v^{\gamma_{2}}-\delta w_{2}, \ &\ \ x\in \Omega, \ t>0 , \end{array} \right. \end{equation*} $

where $ \Omega $ is a bounded smooth domain of $ \mathbb{R}^{n}(n\geq 1), $ the parameters $ a, b, \alpha, \beta, \gamma, \delta, \gamma_{1}, \gamma_{2} > 0, \kappa > 1, $ and nonnegative functions $ \phi(\varrho) = (\varrho+1)^{m}, $ $ \varphi(\varrho) = \chi \varrho(\varrho+1)^{\theta-1} $ and $ \psi(\varrho) = \xi \varrho(\varrho+1)^{l-1} $ for $ \varrho\geq 0 $ with $ m, \theta, l \in \mathbb{R} $ and $ \chi, \xi > 0. $ In the present work, we improve the boundedness criteria established in previous work and further show that under the corresponding critical cases, namely, assume that $ \theta+\gamma_{1} = \max\{l+\gamma_{2}, \kappa\}\geq m+\frac{2}{n}+1 $ with $ m > -\frac{2}{n}, n\geq 3, $ if one of the following conditions holds:

(a) when $ \theta+\gamma_{1} = l+\gamma_{2} = \kappa, $ if $ \theta\geq l\geq 1 $ and $ \frac{[(\kappa-1-m)n-2](2\alpha \chi-\gamma\xi)}{2(l-1)+(\kappa-1-m)n} = b, $ or $ l\geq \theta\geq 1 $ and $ \frac{2\alpha\chi[(\kappa-1-m)n-2]} {2(\theta-1)+(\kappa-1-m)n} = b; $

(b) when $ \theta+\gamma_{1} = \kappa > l+\gamma_{2}, $ if $ \theta\geq 1 $ and $ \frac{2\alpha \chi[(\kappa-1-m)n-2]}{2(\theta-1)+(\kappa-1-m)n} = b, $

then the system still possesses at least a global classical solution, which is bounded in $ \Omega \times (0, \infty) $. Additionally, we have also explored the long time behavior of the classical solution mentioned above.