  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
