具有矩阵值灵敏度的二维Keller-Segel系统解的全局存在性

Global existence in a 2D Keller-Segel system with rotational sensitivity

摘要: 本文研究具有矩阵值灵敏度的 Keller-Segel 方程组
\begincasesu_t=\Delta u-\nabla \cdot\left(u S_\theta \nabla v\right), & x \in \Omega, t>0, \\ v_t=\Delta v-v+u, & x \in \Omega, t>0\endcases
的初边值问题, 其中u满足零通量边界条件, v满足齐次 Neumann 边界条件,Ω=B_R(0)⊂\mathbbR²是以原点为中心、R为半径的圆盘,S_\theta=\left(\beginarraycc\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\endarray\right)为旋转矩阵, θ∈(－π/2, π/2). 本文证明: 若非负径向初值u₀满足\int_\Omega u_0(x) \mathrmdx < \frac8\pi\cos\theta,则该系统存在唯一的全局经典解.

Abstract: This paper investigates the initial-boundary value problem of the Keller-Segel system with matrix sensitivity
\begincasesu_t=\Delta u-\nabla \cdot\left(u S_\theta \nabla v\right), & x \in \Omega, t>0, \\ v_t=\Delta v-v+u, & x \in \Omega, t>0\endcases
Whereuis subject to the homogeneous Neumann boundary condition, vsatisfies the same Neumann boundary condition, and Ω=B_R(0)⊂\mathbbR² denotes the disc of radius Rcentered at the origin. The matrix S_\theta=\left(\beginarraycc\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\endarray\right),θ∈(－π/2, π/2), represents thematrix sensitivity. We prove that if the nonnegative radially symmetric initial data u₀ satisfy \int_\Omega u_0(x) \mathrmdx < \frac8\pi\cos\theta, then the system admits a unique global classical solution.