Absolute instabilities of travelling wave solutions in a Keller–Segel model. (11th October 2017)

• Record Type: Journal Article
• Authors: Davis, P N; van Heijster, P; Marangell, R
• Is Part Of: Nonlinearity. Volume 30:Number 11(2017:Nov.)
• Page Start: 4029
• Page End: 4061
• Publication Date: 2017-10-11
• Subjects: Keller–Segel model -- logarithmic chemosensitivity -- travelling wave solutions -- spectral stability -- weighted essential spectrum -- absolute instabilities -- absolute spectrum; 35C07 -- 35B35 -- 35K59 -- 47A25 -- 47A75 -- 92C17; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa842f
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We investigate the spectral stability of travelling wave solutions in a Keller–Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear, and linear consumption rate. Linearising around the travelling wave solutions, we locate the essential and absolute spectrum of the associated linear operators and find that all travelling wave solutions have parts of the essential spectrum in the right half plane. However, we show that in the case of constant or sublinear consumption there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. For the constant and sublinear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. We observe that this crossing always occurs off of the real axis.