  This paper deals with the existence of traveling waves type patterns in the
case of the Keller-Segel model with logarithmic sensitivity. The cases in which
the diffusion is linear and nonlinear with flux-saturated (of the relativistic
heat equation-type) are fully analyzed by comparing the difference between both
cases. Moreover, special attention is paid to traveling waves with compact
support or with support in the semi-straight line. The existence of these
patterns is rigorously proved and the differences between both cases (linear or
nonlinear diffusion) are analyzed.
