具奇异敏感及Logistic源的趋化系统解的整体存在性

Global Existence of Solutions for a Chemotaxis System with Logistic Source and Singular Sensitivity

摘要: 本文研究具有奇异敏感性和logistic源的趋化系统：$u_t=\Delta u+\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right)+ru-\mu u^k$ ，$v_t=\Delta v-v+uv$ ，其中$\Omega \in R^n\left(n\ge 2\right)$ 是一个光滑有界的凸区域，$\chi ,r,\mu >0$ 且$k\ge 2$ 。在$\chi <2$ 的条件下，当$k=2$ ，$\mu >\frac{n{\chi }^2}{8-4\chi }$ 或者$k>2$ 时，系统存在整体古典解。

Abstract: This paper deal with the chemotaxis systems with singular sensitivity and logistic source: $u_t=\Delta u+\chi \nabla \cdot \left(\frac{u}{v}\nabla v\right)+ru-\mu u^k$ , $v_t=\Delta v-v+uv$ , where $\Omega \in R^n\left(n\ge 2\right)$ is a smooth bounded convex domain, $\chi ,r,\mu >0$ and $k\ge 2$ . If $\chi <2$ and $k=2$ , $\mu >\frac{n{\chi }^2}{8-4\chi }$ or $k>2$ , then the system admits global classical solutions.