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

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

2012-02-17

arxiv.org/abs/1202.3880v1

Mathematics

Analysis of PDEs

This preprint version. Published in Advances in Differential Equations 16 (2011) 31-60

Scientific paper

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.

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