Critical blow-up curve in a two-species chemotaxis system with two chemicals involving flux-limitation

Abstract: We investigate the following two-species chemotaxis system with two chemicals involving flux-limitation \begin{align}\tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot \left(u(1+|\nabla v|^2)^{-\frac{p}{2}}\nabla v\right), & x \in Ω, \ t > 0, \ 0 = Δv - μw + w, \quad μ{w}=f_Ω w, & x \in Ω, \ t > 0, \ w_t = Δw - \nabla \cdot \left(w (1+|\nabla z|^2)^{-\frac{q}{2}} \nabla z\right), & x \in Ω, \ t > 0, \ 0 = Δz - μu + u, \quad μ{u}=f_Ω u, & 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 $p,q \in \mathbb{R}$ and $Ω\subset \mathbb{R}^n$ is a smooth bounded domain. In this paper, we identify a critical blow-up curve ( i.e $p=\frac{n-2}{n-1}$ and $q=\frac{n-2}{n-1}$ in the square $(0,\frac{n-2}{n-1}] \times (0,\frac{n-2}{n-1}]$) for system ($\star$) with $n\geq 3$ and $p,q>0$. Specifically, \begin{itemize} \item when $Ω=B_R(0) \subset \mathbb{R}^n$ with $n\geq 3$, if $0<p<\frac{n-2}{n-1}$ and $0<q<\frac{n-2}{n-1}$, there exist radially symmetric initial data such that the corresponding solution blows up in finite time; \item for any general smooth bounded domain, if either $n=1$ ( with $p,q \in \mathbb{R}$ arbitrary) or $n\geq 2$ with $p>\frac{n-2}{n-1}$ or $q>\frac{n-2}{n-1}$, then solutions exist globally and remain bounded. \end{itemize}