Abstract: We review results on global existence and long time behavior of solutions for PDE models of chemotaxis. In particular, we present the existence and the stability of large-amplitude traveling wave solutions to a system of nonlinear conservation laws
derived from the classical Keller-Segel model for the repulsive chemotaxis. While in the attractive chemotaxis case, we found oscillatory traveling wave solutions which are biologically relevant. Results on initial boundary value problems will also be presented.
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