带Logistic源的奇异趋化系统解的整体存在性

Global Existence of Classical Solutions to a Singular Chemotaxis System with Logistic Source

摘要: 本文研究一类在齐次Neumann边界条件下的具有奇异灵敏度和Logistic源的抛物–抛物趋化系统：$u_t=\Delta u-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right)+ru-\mu u^k$ ，$v_t=\Delta v-v+u$ ，其中$\Omega \subset {ℝ}^n$ 为光滑有界凸域，$\mu ,\chi >0$ ，$r\in ℝ$ 。证明了当于$k>1$ 及$\chi \le \frac{4}{n}$ 时，系统存在唯一的整体古典解。

Abstract: This paper investigates a class of parabolic chemotaxis systems with singular sensitivity and Logistic sources under homogeneous Neumann boundary conditions: $u_t=\Delta u-\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right)+ru-\mu u^k$ , $v_t=\Delta v-v+u$ , where $\Omega \subset {ℝ}^n$ is a smooth bounded convex domain, $\mu ,\chi >0$ , $r\in ℝ$ . It is proved that for $k>1$ with $\chi \le \frac{4}{n}$ , the system admits a unique global classical solution.