Dynamics of some partial differential equation models arising in fluid mechanics and biology

Alhasanat, Ahmad Salman (2017) Dynamics of some partial differential equation models arising in fluid mechanics and biology. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this thesis, we study the dynamics of some partial differential models arising in fluid mechanics and biology. First, we analyze a long-wave model for a liquid thin film on an inclined periodic substrate that is valid at a near-critical Reynolds number. The existence and the uniqueness, as well as the asymptotic formula, of a periodic steady-state are derived. Floquet-Bloch theory and asymptotic analysis are carried out to study the stability in a weighted functional space. The generalized Burgers equation is another fluid model that we consider. After transforming the problem into a constant coefficients problem, a shooting method is used to prove the existence of separable solutions. The total number of them is given and the uniqueness of the positive solution is proved. The stability of the small-amplitude positive steady-state is provided using the bifurcation analysis. Dynamics of a two-species competition model with diffusion is studied in the last part. The minimal wave speed selection mechanism (linear vs. nonlinear) is investigated. Hosono conjectured that there is a critical value of the birth rate so that the speed selection changes only at this value. We prove a modified version of this conjecture and establish some new results for the linear and the nonlinear speed selection. The local and the global stability, using the comparison principle together with the squeezing technique, of the traveling wavefront are studied in a weighted functional space. Some open problems and future works are presented.

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