Local well posedness and instability of travelling waves in a chemotaxis model (Q536518)

An important mechanism in biology discovered in the context of stem cells and neurons is chemotaxis, the directed movement of a cell species towards the gradient of a chemical. The Keller-Segel model for chemotaxis with a nonlinear diffusion coefficient, and a singular sensitivity function is considered and the authors 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. Part of the essential spectrum of the linearization, which has unbounded coefficients on one half-axis, is determined. By generalizing the principle of linearized instability without spectral gap to fully nonlinear parabolic problems, nonlinear instability of the waves in certain cases is obtained.