Chemotaxis can prevent thresholds on population density

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+κu-μu^2$ $0 =Δv-v+u$ $\partial_νv|_{\partialΩ} = \partial_νu|_{\partialΩ}=0,$ $u(0,\cdot)=u_0$ 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

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