Kinks and solitons in linear and nonlinear-diffusion Keller-Segel type models with logarithmic sensitivity

This paper investigates the existence of traveling–wave–type patterns in the Keller–Segel model with logarithmic sensitivity. We consider both the linear diffusion case and the nonlinear, flux-saturated diffusion of relativistic heat–equation type, providing a detailed comparison between the two regimes. Particular attention is devoted to traveling waves exhibiting compact support or support restricted to a half-line. We rigorously establish the existence of such patterns and highlight the qualitative differences arising from the choice of diffusion mechanism.