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

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

$$\{\begin{array}{ll}{v}_t=\mathrm{\nabla }\cdot (\varphi (v)\mathrm{\nabla }v-\phi (v)\mathrm{\nabla }{w}_1+\psi (v)\mathrm{\nabla }{w}_2)+av-b{v}^{\kappa },& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }{w}_1+\alpha v^{{\gamma }_1}-\beta w_1,& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }{w}_2+\gamma v^{{\gamma }_2}-\delta w_2,& x\in \mathrm{\Omega },t>0,\end{array}$$

where $\mathrm{\Omega }$ is a bounded smooth domain of ${\mathbb{R}}^n(n\ge 1),$ the parameters $a,b,\alpha ,\beta ,\gamma ,\delta ,{\gamma }_1,{\gamma }_2>0,\kappa >1,$ and nonnegative functions $\varphi (\varrho )=(\varrho +1{)}^m,$ $\phi (\varrho )=\chi \varrho (\varrho +1{)}^{\theta -1}$ and $\psi (\varrho )=\xi \varrho (\varrho +1{)}^{l-1}$ for $\varrho \ge 0$ 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 $\theta +{\gamma }_1=max\{l+{\gamma }_2,\kappa \}\ge m+\frac{2}{n}+1$ with $m>-\frac{2}{n},n\ge 3,$ if one of the following conditions holds:

(a) when $\theta +{\gamma }_1=l+{\gamma }_2=\kappa ,$ if $\theta \ge l\ge 1$ and $\frac{[(\kappa -1-m)n-2](2\alpha \chi -\gamma \xi )}{2(l-1)+(\kappa -1-m)n}=b,$ or $l\ge \theta \ge 1$ 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 $\theta \ge 1$ 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 $\mathrm{\Omega }\times (0,\mathrm{\infty })$. Additionally, we have also explored the long time behavior of the classical solution mentioned above.