Existence and stability of traveling waves to conservation laws arising from chemotaxis

Abstract

This thesis is concerned with the system of conservation laws transformed via a Cole-Hopf transformation from the Keller-Segel model to repulsive chemotaxis which has various applications, such as biological systems that build transport networks (reinforced random walks [27,28]). Traveling wave solutions to conservation laws arising from chemotaxis with positive chemotactic coeffcient (denoted by X > 0) have been widely studied, while the analysis of traveling waves to conservation laws arising from repulsive chemotaxis (denoted by X < 0) still remains open. The purpose of this thesis will be to develop some existence and stability theories for such a system of conservation laws to repulsive chemotaxis. We first show the existence of travaling waves by using the phase plane analysis, where there are three heteroclinic connections in the system. Numerical simulations will be presented to demonstrate the process of wave propagation. We then proceed to prove the stability of traveling waves by employing the method of weighted energy estimates. We get estimates on the types of integrals which are usually interpreted in terms of energies related to the physical problems behind partial differential equations. This thesis develops the first theoretical results on traveling waves to conservation laws arising from repulsive chemotaxis and more open problems relate to the system will be proposed for future studies.