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

TL;DR

This paper determines a critical blow-up curve for a two-species chemotaxis system with flux-limitation, identifying conditions under which solutions blow up or remain bounded in higher dimensions.

Contribution

It establishes a precise critical curve involving parameters p and q that delineates blow-up and global existence regimes for the system.

Findings

Solutions blow up in finite time when p,q are below the critical curve.

Solutions are globally bounded when p,q exceed the critical curve.

The critical curve is explicitly identified for dimensions n ≥ 3.

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.