Graduation Year

2016

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Yuncheng You, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Committee Member

Sherwin Kouchekian, Ph.D.

Committee Member

Marcus McWaters, Ph.D.

Keywords

Boissonade equations, Global Attractor, Random attractor, Stochastic Brusselator equations

Abstract

The dissertation studies about the existence of three different types of attractors of three multi-component reaction-diffusion systems. These reaction-diffusion systems play important role in both chemical kinetics and biological pattern formation in the fast-growing area of mathematical biology.

In Chapter 2, we prove the existence of a global attractor and an exponential attractor for the solution semiflow of a reaction-diffusion system called Boissonade equations in the $L^2$ phase space. We show that the global attractor is an $(H, E)$ global attractor with the $L^\infty$ and $H^2$ regularity and that the Hausdorff dimension and the fractal dimension of the global attractor are finite. The existence of exponential attractor is also shown. The upper-semicontinuity of the global attractors with respect to the reverse reaction rate coefficient is proved.

In Chapter 3, the existence of a pullback attractor for non-autonomous reversible Selkov equations in the product $L^2$ phase space is proved. The method of grouping and rescaling estimation is used to prove that the $L^4$-norm and $L^6$-norm of solution trajectories are asymptotic bounded. The new feature of pinpointing a middle time in the process turns out to be crucial to deal with the challenge in proving pullback asymptotic compactness of this typical non-autonomous reaction-diffusion system.

In Chapter 4, asymptotical dynamics of stochastic Brusselator equations with multiplicative noise is investigated. The existence of a random attractor is proved via the exponential transformation of Ornstein-Uhlenbeck process and some challenging estimates. The proof of pullback asymptotic compactness here is more rigorous through the bootstrap pullback estimation than a non-dynamical substitution of Brownian motion by its backward translation. It is also shown that the random attractor has the $L^2$ to $H^1$ attracting regularity by the flattening method.

Share

COinS