### Local and global well-posedness for a class of nonlinear dispersive equations

#### Abstract

We study the local and global well-posedness of the initial-value problem for the class of nonlinear dispersive PDEs of the form $u_t - M u_x + F(u)_x = 0, \;\;\; t\in \mathbb{R},$ where $u=u(x,t)$, $x\in \mathbb{R}$ or $x\in \mathbb{T}$. Here $M$ is a linear operator, given in the Fourier space by the multiplication operator: $\hat{Mv} ( \xi ) = |\xi |^{2\beta } \hat{v} (\xi ),$ $\beta \ge \frac12$ and $F$ is a nonlinear, (sufficiently) smooth function. This equation is a generalization of the Korteweg--de Vries (KdV) equation ($\beta = 1$), the Benjamin--Ono (BO) equation ($\beta = \frac12$) and the fifth-order KdV equation ($\beta = 2$). 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 $(F'(r))_{+}$ grows at most like $\left| r\right| ^p$ as $r\to\infty$, for some $p < 4\beta$. 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., $x\in\mathbb{T}$), but they are also valid in the real-line case (when $x\in \mathbb{R} )$.

#### Article information

Source
Adv. Differential Equations, Volume 9, Number 1-2 (2004), 85-132.

Dates
First available in Project Euclid: 18 December 2012