Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity

Abstract

We investigate the parabolic-elliptic Keller-Segel model \begin{align*}\left\{\begin{array}{r@{\,}l@{\quad}l@{\quad}l@{\,}c} u_{t}&=Δu-\,χ\nabla\!\cdot(\frac{u}{v}\nabla v),\ &x\inΩ,& t>0,\\ 0&=Δv-\,v+u,\ &x\inΩ,& t>0,\\ \frac{\partial u}{\partialν}&=\frac{\partial v}{\partialν}=0,\ &x\in\partialΩ,& t>0,\\ u(&x,0)=u_0(x),\ &x\inΩ,& \end{array}\right. \end{align*} in a bounded domain $\Omega \subset {\mathbb{R}}^n$ $(n\ge 2)$ with smooth boundary. \noindent We introduce a notion of generalized solvability which is consistent with the classical solution concept, and we show that whenever $0<\chi <\frac{n}{n-2}$ and the initial data satisfy only certain requirements on regularity and on positivity, one can find at least one global generalized solution.

19 pages; corrected typos in the metadata

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