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Critical blow-up lines in a two-species quasilinear chemotaxis system with two chemicals

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 Ω, \ t > 0, \\ 0 = Δv - μ_w + w, \quad μ_w=\fint_Ωw, & x \in Ω, \ t > 0, \\ w_t = Δw - \nabla \cdot \left(w \nabla z\right), & x \in Ω, \ t > 0, \\ 0 = Δz - μ_u + u, \quad μ_u=\fint_Ωu, & x \in Ω, \ t > 0, \\ \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = \frac{\partial w}{\partial ν} = \frac{\partial z}{\partial ν} = 0, & x \in \partial Ω, \ t > 0, \\ u(x, 0) = u_0(x), \quad w(x, 0) = w_0(x), & x \in Ω, \end{cases} \end{align} where $Ω\subset \mathbb{R}^n$ ($n \geq3$) 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 $Ω$ 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 $Ω\subset \mathbb{R}^n$, if $q-p<2-\frac{n}{2}$, all solutions are globally bounded; \item for any general smooth bounded domain $Ω\subset \mathbb{R}^n$, 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 ($\star$).