Advances in Differential Equations
- Adv. Differential Equations
- Volume 16, Number 1/2 (2011), 31-60.
Local well posedness and instability of travelling waves in a chemotaxis model
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.
Adv. Differential Equations Volume 16, Number 1/2 (2011), 31-60.
First available in Project Euclid: 18 December 2012
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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