UC Santa Barbara

In recent years there has been an intense activity in the study of harmonic analysis and its application to partial differential equations (PDEs). The tools of harmonic analysis assist in the discovery of important properties of certain PDEs; amid these PDEs one finds the study of nonlinear dispersive equations.

In particular, the problems of establishing local and global well-posedness under minimal regularity requirement of the given data, the long-time behavior of local solutions to these models, scattering, blow-up, and the unique continuation properties of the solutions have been extensively studied.

Among the systems considered one finds the Korteweg-de Vries equation, the Schrödinger system and the Benjamin-Ono equation, all occurring in different physical problems, mainly nonlinear wave propagation. Furthermore, under certain circumstances, these all admit solitary wave solutions called traveling waves, which have important applications in fiber optics, magnetics and genetics.

In this thesis we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin-Ono equation given in weighted Sobolev spaces. More precisely, we prove that the uniqueness property based on a decay requirement at three times cannot be lowered to two times even by imposing stronger decay on the initial data.