This paper deals with the initial-boundary value problem for the rotational chemotaxis system with two species and two chemicals with zero-flux boundary condition for u, w and zero-Neumann boundary condition for v, z, where \(\Omega \) is a bounded domain in \(\mathbb {R}^2\) with smooth boundary \(\partial \Omega \) , \(\begin{aligned} S_{\theta } = \Big [ \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \Big ] \end{aligned}\) is a rotation matrix with \(\theta \in (-\frac{\pi }{2}, \frac{\pi }{2})\) and \(\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta >0\) . Let \(m_1,m_2\) be the initial mass for u and w respectively. We show that: If \(m_1m_2- \frac{4\pi }{\cos \theta }(\frac{m_1}{\chi \beta }+\frac{m_2}{\xi \delta })>0\) , then there exists finite-time blow-up solution to the system ( \(\star \) );
If \(m_1m_2- \frac{2\pi }{\cos \theta }(\frac{m_1}{\chi \beta }+\frac{m_2}{\xi \delta })>0\) and \(\partial \Omega \) contains a line segment, then there exists finite-time blow-up solution to the system ( \(\star \) ).
We point out that if \(\Omega \) is a disc in \(\mathbb {R}^2\) and \(m_1m_2- \frac{2\pi }{\cos \theta }(\frac{m_1}{\chi \beta }+\frac{m_2}{\xi \delta })<0\) , it is easy to show that the global existence of the solution to the system ( \(\star \) ) using the energy functional method. Therefore, the line \(m_1m_2- \frac{2\pi }{\cos \theta }(\frac{m_1}{\chi \beta }+\frac{m_2}{\xi \delta })=0\) is critical in this sense.
