In this study, we explore the quasilinear two-species chemotaxis system with two chemicals \begin{ align}\tag{ $\star$} \begin{ cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot \left(S(u) \nabla v\right), & x \in \Omega, \ t > 0, \\ 0 = \Delta v - \mu_w + w, \quad \mu_w=\fint_{ \Omega}w, & x \in \Omega, \ t > 0, \\ w_t = \Delta w - \nabla \cdot \left(w \nabla z\right), & x \in \Omega, \ t > 0, \\ 0 = \Delta z - \mu_u + u, \quad \mu_u=\fint_{ \Omega}u, & x \in \Omega, \ t > 0, \\ \frac{ \partial u}{ \partial \nu} = \frac{ \partial v}{ \partial \nu} = \frac{ \partial w}{ \partial \nu} = \frac{ \partial z}{ \partial \nu} = 0, & x \in \partial \Omega, \ t > 0, \\ u(x, 0) = u_0(x), \quad w(x, 0) = w_0(x), & x \in \Omega, \end{ cases} \end{ align} where $\Omega \subset \mathbb{ R}^n$ ($n \geq3$) is a smooth bounded domain. The functions $D(s)$ and $S(s)$ exhibit asymptotic behavior of the form \begin{ align*} D(s) \simeq k_D s^p \ \text { and} \ S(s) \simeq k_S s^q, \quad s \gg 1 \end{ align*} with $p,q \in \mathbb{ R}$. We prove that \begin{ itemize} \item when $\Omega$ is a ball, if $q-p>2-\frac{ n}{ 2}$ and $q>1-\frac{ n}{ 2}$, there exist radially symmetric initial data $u_0$ and $w_0$, such that the corresponding solutions blow up in finite time; \item for any general smooth bounded domain $\Omega \subset \mathbb{ R}^n$, if $q-p<2-\frac{ n}{ 2}$, all solutions are globally bounded; \item for any general smooth bounded domain $\Omega \subset \mathbb{ R}^n$, if $q<1-\frac{ n}{ 2}$, all solutions are global. \end{ itemize} We point out that our results implies that the system ($\star$) possess two critical lines $ q-p=2-\frac{ n}{ 2}$ and $q=1-\frac{ n}{ 2}$ to classify three dynamics among global boundedness, finite-time blow-up, and global existence of solutions to system ($\star$).
Categories: math.AP