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 { u t = Δ u − ∇ ⋅ ( u S θ ∇ v ) , x ∈ Ω , t > 0 , 0 = Δ v − 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 R 2 with smooth boundary ∂ Ω , S θ = [ cos θ − sin θ sin θ cos θ ] , is a rotation matrix with θ ∈ ( − π 2 , π 2 ) . We show that: Let Ω ⊂ R 2 be a general smooth bounded domain. If m > 8 π cos θ , then there exists nonnegative initial data u 0 satisfying ∫ Ω u 0 d x = m , such that the corresponding nonradial solution of system (*) blows up in finite time and the blow-up point lies in Ω. If m > 4 π cos θ and ∂ Ω contains a line segment, then there exists nonnegative initial data u 0 satisfying ∫ Ω u 0 d x = 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 = B R ( 0 ) be a disc in R 2 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 ∫ Ω u 0 d x < 8 π cos θ , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

“…Concerning the case ℝ 2 , the authors in [8] proved that there exists a threshold value for the initial mass 𝑀 = ∫ 𝑢 0 𝑑𝑥 that relates to the existence and blow up of solutions: if 𝑀 < 8𝜋∕𝜅, then solutions exist globally and if 𝑀 > 8𝜋∕𝜅, then solutions blow up in a finite time. For a bounded domain with smooth boundary in ℝ 2 , Li and Wang established the finite-time blow-up and boundedness for system (1.1) in [31]. On the other hand for the case ℍ 2 , Pierfelice and Maheux obtained the local and global well-posedness results under the sub-critical condition and a blow-up result in [32].…”