Global well-posedness for a class of dispersive equations

This dissertation is concerned with the study of the global well-posedness of the initial value problem for the class of nonlinear dispersive PDEs of the form ut-Mux+Fu x=0,t∈R , where $u=ux,t,x\in R$ or $x\in T$. Here M is a linear operator, given in the Fourier space by the multiplication operator: , $b\ge 12$ and F is a nonlinear (sufficiently) smooth function. This equation is a generalization of the Korteweg-de Vries (KdV) equation ($b$ = 1), the Benjamin-Ono (BO) equation ($b=12$) and the fifth-order KdV equation ($b$ = 2). The nonlinearity can be very general, but a certain growth condition must be imposed in order to obtain global results. Roughly speaking, we impose that grows at most like $rp$ as $r\to \infty$, for some . Global existence of solutions is, therefore, intimately related to the balance between the strength of the nonlinearity and the dispersion relation. The semigroup methods developed by Goldstein-Oharu-Takahashi are being successfully applied here. Most of the results are presented in the periodic case (i.e. $x\in T$), but they are also valid in the real line case (when $x\in R$).