| Contents |
| 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_\nu v&=\partial_\nu u=0 , u(0,\cdot)=u_0,
\end{align*}
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.\\
{\bf Keywords: }chemotaxis, logistic source, blow-up, hyperbolic-elliptic system |
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