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

Abstract

This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation with zero-flux boundary condition for u and zero-Neumann boundary condition for v, where Ω is a bounded domain in with smooth boundary , is a rotation matrix with . We show that:Let be a general smooth bounded domain.If , then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω.If and contains a line segment, then there exists nonnegative initial data u 0 satisfying , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies on the line segment of .Let be a disc in 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 , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.