Summary: This paper deals with the initial-boundary value problem for a Keller-Segel system with rotation \[ \begin{cases} u_t=\Delta u-\nabla\cdot(uS_\theta\nabla\nu),\quad & x\in\Omega,t>0,\\ 0=\Delta\nu-\nu+u,\quad & x\in\Omega,t>0, \end{cases}\tag{\(\star\)} \] with zero-flux boundary condition for \(u\) and zero-Neumann boundary condition for \(v\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) with smooth boundary \(\partial\Omega\), \[ S_\theta=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}, \] is a rotation matrix with \(\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). We show that:

●

Let \(\Omega\subset\mathbb{R}^2\) be a general smooth bounded domain.

*

If \(m>\frac{8\pi}{\cos\theta}\), then there exists nonnegative initial data \(u_0\) satisfying \(\int_\Omega u_0\mathrm{d}x=m\), such that the corresponding nonradial solution of system \((\star)\) blows up in finite time and the blow-up point lies in \(\Omega \).

*

If \(m>\frac{4\pi}{\cos\theta}\) and \(\partial\Omega\) contains a line segment, then there exists nonnegative initial data \(u_0\) satisfying \(\int_\Omega u_0\mathrm{d}x=m\), such that the corresponding nonradial solution of system \((\star)\) blows up in finite time and the blow-up point lies on the line segment of \(\partial\Omega\).

●

Let \(\Omega=B_R(0)\) be a disc in \(\mathbb{R}^2\) with radius \(R>0\) centered at origin. Although there is a rotation effect in system \((\star)\), solutions still preserve radial symmetry of initial data. If nonnegative radially symmetric initial data \(u_0\) satisfies \(\int_\Omega u_0\mathrm{d}x<\frac{8\pi}{\cos\theta}\), then the corresponding radial solution of system \((\star)\) exists globally in time and is globally bounded.