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 = \Delta u - \nabla \cdot \left(u(1+|\nabla v|^2)^{-\frac{p}{2}}\nabla v\right), & x \in \Omega, \ t > 0, \\ 0 = \Delta v - \mu_w + w, \quad \mu_{w}=f_{\Omega} w, & x \in \Omega, \ t > 0, \\ w_t = \Delta w - \nabla \cdot \left(w (1+|\nabla z|^2)^{-\frac{q}{2}} \nabla z\right), & x \in \Omega, \ t > 0, \\ 0 = \Delta z - \mu_u + u, \quad \mu_{u}=f_{\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 $p,q \in \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. In this paper, we identify a critical blow-up curve for system ($\star$) with $n\geq 3$. If $p<\frac{n-2}{n-1}$ and $q<\frac{n-2}{n-1}$, and $\Omega=B_R(0) \subset \mathbb{R}^n$ with $n\geq 3$, there exist radially symmetric initial data such that the corresponding solution blows up in finite time; if either $p>\frac{n-2}{n-1}$ or $q>\frac{n-2}{n-1}$ with $n\geq 2$, then solutions exist globally and remain bounded.