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

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