Jisun Lim Abstract
PhD, August 2007The Qualitative Study of a Chemical Reaction Diffusion System and Some Integral Equations
Advisor: Congming Li
This dissertation contributes to the theoretical study of nonlinear differential equations and integral equations. First, we present mass balance models and analyze the stability of two systems of reaction-diffusion equations that describes two different reactions kinetics. Two models that we deal with are related to one reversible reaction and to bicarbonate reactions in blood. The Lyapunov's method is utilized to prove that such models are stable as time proceeds. Additionally, the equilibrium concentrations for two systems are calculated explicitly. On the bicarbonate system, we find that the system is stable assuming that any reaction rate constant is small enough. Second, we explore the qualitative properties of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality. The properties that we examine are regularity and asymptotic behavior of solutions of a set of Euler-Lagrange integral equations. A new technique for regularity problems is developed. It has been shown useful to establish the optimal integrability of the solutions of integral equations. Based on the optimal integrability of the solutions, we investigate the asymptotic behaviors of solutions of a system of integral equations. The decay rate and blow-up rate of the solutions are explicitly calculated in the form of certain integrals.
