A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data

TL;DR

This paper introduces a new generalized solution framework for the Keller-Segel chemotaxis system with logarithmic sensitivity, extending global solvability results to larger parameter ranges and nonradial initial data.

Contribution

A novel generalized solution concept is developed, allowing for extended parameter ranges and nonradial data in the Keller-Segel system with logarithmic sensitivity.

Findings

Global solutions exist for a broader range of the chemotactic sensitivity parameter  under the new framework.

The solutions are well-defined for large, nonradial initial data with certain regularity and positivity.

The solutions have local integrability properties in space-time.

Abstract

The chemotaxis system \[ \left\{ \begin{array}{l} u_t = \Delta u - \chi\nabla \cdot (\frac{u}{v}\nabla v), v_t=\Delta v - v+u, \end{array} \right. \] is considered in a bounded domain $\Omega\subset \mathbb{R}^n$ with smooth boundary, where $\chi>0$. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $\chi$ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that\[ \chi < \left\{ \begin{array}{ll} \infty \qquad & \mbox{if } n=2, \sqrt{8} \qquad & \mbox{if } n=3, \frac{n}{n-2} \qquad & \mbox{if } n\ge 4, \end{array} \right. \] for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that \[ u\in L^1_{loc}(\bar\Omega\times [0,\infty)). \]