2016

Ph.D.

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Yuncheng You, 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.

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