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}& = \Delta u-\, \chiabla\!\cdot(\frac{u}{v}abla v), \ &x\in\Omega, & t>0, \\ 0& = \Delta v-\, v+u, \ &x\in\Omega, & t>0, \\ \frac{\partial u}{\partialu}& = \frac{\partial v}{\partialu} = 0, &x\in\partial \Omega, & t>0, \\ u(&x, 0) = u_0(x), \ &x\in\Omega, & \end{array}\right. \end{align*} $in a bounded domain $ \Omega\subset\mathbb{R}^n $ $ (n\geq2) $ with smooth boundary.We introduce a notion of generalized solvability which is consistent with the classical solution concept, and we show that whenever $ 0