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

2012/02/17 by Meyries, Martin
#35B40 #35G25 #35K55 #47D06 #47E05 #92B05 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1202.3880

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.

Related

• Spectral Stability of Travelling Wave Solutions in a Keller-Segel Model
• Traveling waves for the Keller-Segel-FKPP equation with strong chemotaxis
• Traveling waves to a logarithmic chemotaxis model with fast diffusion and singularities
• Global well-posedness of large perturbations of traveling waves in a\n hyperbolic-parabolic system arising from a chemotaxis model
• Existence and Instability of Traveling Pulses of Keller-Segel System with Nonlinear Chemical Gradients and Small Diffusions