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

2022; IOP Publishing; Volume: 36; Issue: 1 Linguagem: Inglês

10.1088/1361-6544/aca3f6

1361-6544

Yuxiang Li, Wanwan Wang,

Microtubule and mitosis dynamics

Abstract This paper deals with the initial-boundary value problem for a Keller–Segel system with rotation <?CDATA \begin{equation} \bigg\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u S_{\theta} \nabla v), & x \in \Omega,\ t\gt0, \\ 0 = \Delta v - v + u, & x \in \Omega,\ t\gt0,&&(*) \end{array} \end{equation}?> { 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 <?CDATA $\mathbb{R}^2$?> R 2 with smooth boundary <?CDATA $\partial\Omega$?> ∂ Ω , <?CDATA \begin{equation*} S_{\theta} = \Bigg[ \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \Bigg], \end{equation*}?> S θ = [ cos θ − sin θ sin θ cos θ ] , is a rotation matrix with <?CDATA $\theta \in (-\frac{\pi}{2}, \frac{\pi}{2})$?> θ ∈ ( − π 2 , π 2 ) . We show that: Let <?CDATA $\Omega \subset \mathbb{R}^2$?> Ω ⊂ R 2 be a general smooth bounded domain. If <?CDATA $m \gt \frac{8\pi}{\cos \theta}$?> m > 8 π cos θ , then there exists nonnegative initial data u 0 satisfying <?CDATA $\int_{\Omega} u_0 \,\mathrm{d} x = m$?> ∫ 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 <?CDATA $m \gt \frac{4\pi}{\cos \theta}$?> m > 4 π cos θ and <?CDATA $\partial \Omega$?> ∂ Ω contains a line segment, then there exists nonnegative initial data u 0 satisfying <?CDATA $\int_{\Omega}u_0\,\mathrm{d}x = m$?> ∫ 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 <?CDATA $\partial\Omega$?> ∂ Ω . Let <?CDATA $\Omega = B_R(0)$?> Ω = B R ( 0 ) be a disc in <?CDATA $\mathbb{R}^2$?> 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 <?CDATA $\int_{\Omega}u_0\,\mathrm{d}x\lt\frac{8\pi}{\cos\theta}$?> ∫ Ω u 0 d x < 8 π cos θ , then the corresponding radial solution of system (*) exists globally in time and is globally bounded.