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Finite-time blow-up in a quasilinear two-species chemotaxis system with two chemicals

This paper investigates the finite-time blow-up phenomena to a quasilinear two-species chemotaxis system with two chemicals \begin{align}\tag{$\star$} \begin{cases} u_t = \nabla \cdot \left(D_1(u) \nabla u\right) - \nabla \cdot \left(u \nabla v\right), & x \in Ω, \ t > 0, 0 = Δv - μ_2 + w, \quad μ_2=\fint_Ωw, & x \in Ω, \ t > 0, w_t = \nabla \cdot \left(D_2(w) \nabla w\right) - \nabla \cdot \left(w \nabla z\right), & x \in Ω, \ t > 0, 0 = Δz - μ_1 + u, \quad μ_1=\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 $Ω\subset \mathbb{R}^n$ $(n \geqslant 3)$ is a smoothly bounded domain. The nonlinear diffusion functions \( D_1(s) \) and \( D_2(s) \) are of the following forms: \begin{align*} D_1(s)\simeq s^{m_1-1} \quad \text{and}\quad D_2(s) \simeq s^{m_2-1}, \quad m_1,m_2> 1 \end{align*} for $s\geqslant 1$. For the classical two-species chemotaxis system with two chemicals (i.e. the second and fourth equations are replaced by $0 = Δv - v + w$ and $0 = Δz - z + u$ ), Zhong [J. Math. Anal. Appl., 500 (2021), Paper No. 125130, pp. 22.] showed that the system possesses a globally bounded classical solution in the case that \[ m_1 + m_2 < \max\left\{m_1m_2 + \frac{2m_1}{ n},\ m_1m_2 + \frac{2m_2 }{ n}\right\}. \] Complementing the boundedness result, we prove that the system ($\star$) admits solutions that blow up in finite time, if \[ m_1 + m_2 > \max\left\{ m_1m_2 + \frac{2m_1}{ n},\ m_1m_2 + \frac{2m_2}{ n}\right\} \] with $n\geqslant 3$.