Chemotaxis can prevent thresholds on population density

Abstract

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t=-\nabla \cdot (u\nabla v)+\kappa u-\mu u^2$ $0=\Delta v-v+u$ ${\partial }_{\nu }v{|}_{\partial \Omega }={\partial }_{\nu }u{|}_{\partial \Omega }=0,$ $u(0,\cdot )=u_0$ in balls in ${\mathbb{R}}^n$, which we then use to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler to the higher dimensional (radially symmetric) case. Keywords: chemotaxis, logistic source, blow-up, hyperbolic-elliptic system

25 pages; removed Sec. 3.2 of version 1 due to an error in the proof. Cor. 10 (formerly Cor. 11) now relies on results from [20] instead; main results and arguments remain unchanged