Date of Award

2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Chair

Shangbing Ai

Committee Member

Wenzhang Huang

Committee Member

Huaming Zhang

Committee Member

Marius Nkashama

Committee Member

Shan Zhao

Subject(s)

Chemotaxis., Traveling-wave tubes., Partial differential equations.

Abstract

In this dissertation, we investigate the existence and stability of traveling wave solutions of a chemotaxis model system, which consists of two reaction-diffusion equations, of which containing the chemotaxis term. We prove the existence of a family of traveling wave solutions, one for each wave speed c > c_* for some c_* > 0. Due to continuity of these wave solutions on c, they are unstable in the Banach space X := C ( R ̅) × C(R ̅), where C(R ̅) is the Banach space, equipped with the uniform norm, of all continuous functions U on R with both U(±∞) being finite. We prove that each of wave solutions is unstable in a weighted space X_α of X with the weight function (1 + e^αξ ), where the waves and their translates are not contained in the space X_α. This is done by establishing the local well-posedness of the corresponding evolution equations in a neighborhood of the wave solution in X_α, and showing the essential spectrum of the linearization about the wave solution intersects the right half-plane. After that we introduce the weighted space X ̃_γ with the weight function e^γξ to shift the essential spectrum of the linear operator at the wave solution to the left half-plane, and then use Evans function, with the diffusion coefficient close to one and the chemotaxis coefficient close to zero, to prove that all eigenvalues are located to the left half-plane, which leads to linear stability of the wave solution. We finally consider a special case when the diffusion coefficients are equal and there is no chemotaxis, and prove nonlinear asymptotic stability of the wave solution in( X) ̃_γ by restricting initial data to both weighted spaces X_α and ( X) ̃_γ. Throughout the dissertation, some well-known results and definitions are included to help provide a more complete picture of the relevant concepts.

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