#### 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.

#### Recommended Citation

Albashaireh, Reem N., "Traveling wave solutions of a chemotaxis model : existence and stability" (2014). *Dissertations*. 52.

https://louis.uah.edu/uah-dissertations/52