## Mathematics Graduate Theses & Dissertations

Spring 1-1-2013

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Congming Li

Stephen Preston

Jeanne N. Clelland

Jeffrey S. Fox

Martin Walter

#### Abstract

This thesis consists of two parts: in part one (Chapter 3, 4, 5), we study the qualitative and quantitative properties of the positive solutions of two types of nonlinear integral systems: Wolff type system and weighted Hardy-Littlewood-Sobolev system; in part two (Chapter 6), we prove the local in time existence of solutions of string equations in R2 with periodic boundary conditions.

The Wolff type system consists of fully nonlinear integral equations involving Wolff potential. We first study the system consists of two integral equations. Under some mild integrability conditions, we prove the optimal integrability of the solution. Then we continue to prove the solutions are bounded and Lipschitz continuous. To prove the Lipschitz continuity, an useful lemma-Regularity lifting with shrinking operator is proposed. After we obtained the regularity results of system of two equations, we investigate the system of m equations. We study the asymptotic behaviour of the solutions at infinity as well as the symmetry of the solutions.

The weighted Hardy-Littlewood-Sobolev system is the "Euler-Lagrange equations" of the weighted Hardy-Littlewood-Sobolev inequality. We prove the asymptotic behaviour of the positive solutions near origin and near infinity. According to the sign of the parameter, we divide our proof into two cases. The negative parameter case need a very subtle estimate of the weighted Lp norm of the solution and using iteration to increase its integrability gradually.

The string equations describe the motion of an inextensible string. This is a second order hyperbolic partial differential equation coupled with an ordinary differential equations. We consider the case where the closed (periodic boundary condition) string moves in a plane and prove the solutions exist locally in time.

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