A lower bound on the blow up time for solutions of a chemotaxis system with nonlinear chemotactic sensitivity. (August 2017)

• Record Type: Journal Article
• Authors: Anderson, Jeffrey R.; Deng, Keng
• Is Part Of: Nonlinear analysis. Volume 159(2017)
• Page Start: 2
• Page End: 9
• Publication Date: 2017-08
• Subjects: 35A01 -- 35B44 -- 35K51 -- 35K59; Nonlinear chemotaxis system -- Blow up time -- Global existence; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.11.018
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In a recent study, a lower bound is established on the blow up time for solutions of a chemotaxis system, with nonlinear chemotactic sensitivity u ( u + 1 ) m − 1, set in the three-dimensional unit ball. Here, u is the density of a cell or organism that produces a chemical, with density v, and moves preferentially toward regions of higher concentration of v according to the flux − ∇ u + χ u ( u + 1 ) m − 1 ∇ v . With χ > 0, v is referred to as a "chemoattractant" and, in the case m = 1, the system reduces to a version of the Keller–Segel model. Solutions that blow up in finite time have been previously established for the system on a ball in R n provided n ≥ 2, m > 2 / n . For technical reasons, the lower bound proven for the blow up time applies in such cases when n = 3 and m ≤ 2 . We extend the analysis and resulting lower bound to such a model in general convex domains, with n ≥ 2 and any m .