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Chemotaxis can prevent thresholds on population density

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  • We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to \begin{align*} u_t = -\nabla \cdot (u \nabla v)+\kappa u-\mu u^2\\ 0 = \Delta v - v + u      \\ \partial_v v|\partial \Omega = \partial_v u|\partial \Omega = 0 ,           u(0,\cdot) = u_0, \end{align*} in balls in $\mathbb{R}^n$. They exist globally in time for $\mu\ge 1$ and, for a certain class of initial data, undergo finite-time blow-up if $\mu<1$.
        We then use this blow-up result to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler [26] to the higher dimensional (radially symmetric) case.
    Mathematics Subject Classification: Primary: 35K55; Secondary: 35B44, 35A01, 35A02, 35Q92, 92C17.

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