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

Tobias Black

doi:10.3934/dcdss.2020007

Article Contents

2020, Volume 13, Issue 2: 119-137. Doi: 10.3934/dcdss.2020007

This issue Previous Article Stabilization in a chemotaxis model for virus infection Next Article Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

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

Tobias Black^1,

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Received: March 31, 2017

Revised: October 31, 2017

Published: February 2020

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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-\, \chi\nabla\!\cdot(\frac{u}{v}\nabla v), \ &x\in\Omega, & t>0, \\ 0& = \Delta v-\, v+u, \ &x\in\Omega, & t>0, \\ \frac{\partial u}{\partial\nu}& = \frac{\partial v}{\partial\nu} = 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<\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.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35D99; Secondary: 35A01, 35Q92, 92C17.

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