All ETDs from UAB

Advisory Committee Chair

Yanni Zeng

Advisory Committee Members

Mubenga N Nkashama

Mark Beasley

Wenzhang Huang

David Halpern

Document Type

Dissertation

Date of Award

2021

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

This dissertation’s work is on asymptotic behavior of a Keller- Segel type chemotaxismodel with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf-Cole transformation. Then, for the transformed system, we study the Green’s function of the corresponding linear system, linearized around a constant equilibrium solution, and the results give us a detailed point-wise description of the Greens function, which, itself, is significant in linear theory. Next, with the results on the Green’s function, we study small data solution to the nonlinear system by making use of Duhamel’s principle. This provides a descriptive asymptotic behavior in a point-wise sense, which also leads to L^1 asymptotic behavior, and this is significant as most of the asymptotic behavior studies on related models are done in L^p(p ≥ 2), therefore not descriptive both in time and space. Lastly, to round up the work on this model, we establish the point-wise asymptotic behavior for the original functions of the model prior to the Hopf-Cole transformation.

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