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   $u_t =-\nabla\cdot(u\nabla v)+\kappa u-\mu u^2$   $ 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 
Johannes Lankeit
math.AP