Advances in Differential Equations

Local well posedness and instability of travelling waves in a chemotaxis model

Martin Meyries

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Abstract

We consider the Keller-Segel model for chemotaxis with a nonlinear diffusion coefficent and a singular sensitivity function. We show the existence of travelling waves for wave speeds above a critical value, and establish local well posedness in exponentially weighted spaces in a neighbourhood of a wave. A part of the essential spectrum of the linearization, which has unbounded coefficients on one half-axis, is determined. Generalizing the principle of linearized instability without spectral gap to fully nonlinear parabolic problems, we obtain nonlinear instability of the waves in certain cases.

Article information

Source
Adv. Differential Equations Volume 16, Number 1/2 (2011), 31-60.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854329

Mathematical Reviews number (MathSciNet)
MR2766893

Zentralblatt MATH identifier
1228.35056

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35G25: Initial value problems for nonlinear higher-order equations 35K55: Nonlinear parabolic equations 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47) 92B05: General biology and biomathematics

Citation

Meyries, Martin. Local well posedness and instability of travelling waves in a chemotaxis model. Adv. Differential Equations 16 (2011), no. 1/2, 31--60. https://projecteuclid.org/euclid.ade/1355854329.


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