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

## TL;DR

This paper establishes the existence of global generalized solutions for a Keller-Segel chemotaxis model with singular sensitivity in bounded domains, under specific conditions on the chemotactic sensitivity parameter and initial data.

## Contribution

It introduces a notion of generalized solutions for the Keller-Segel system and proves their existence for a range of sensitivity parameters and initial data.

## Key findings

- Existence of at least one global generalized solution for 0<χ<n/(n-2).
- Generalized solutions are consistent with classical solutions when regularity conditions are met.
- The results extend the understanding of chemotaxis models with singular sensitivities.

## 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. \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.