Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation

This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation {ut=Δu−∇⋅(uSθ∇v),x∈Ω, t>0,0=Δv−v+u,x∈Ω, t>0, with zero-flux boundary condition for u and zero-Neumann boundary condition for v, where Ω is a bounded domain in R2 with smooth boundary ∂Ω , Sθ=[cosθ−sinθsinθcosθ], is a rotation matrix with θ∈(−π2,π2) . We show that: Let Ω⊂R2 be a general smooth bounded domain. If m>8πcosθ , then there exists nonnegative initial data u 0 satisfying ∫Ωu0dx=m , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω. If m>4πcosθ and ∂Ω contains a line segment, then there exists nonnegative initial data u 0 satisfying ∫Ωu0dx=m , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies on the line segment of ∂Ω . Let Ω=BR(0) be a disc in R2 with radius R > 0 centered at origin. Although there is a rotation effect in system (*), solutions still preserve radial symmetry of initial data. If nonnegative radially symmetric initial data u 0 satisfies ∫Ωu0dx<8πcosθ , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.