Differential and Integral Equations

The Cauchy problem for a nonlocal perturbation of the KdV equation

Borys Alvarez Samaniego

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $\mathcal H$ denote the Hilbert transform and $\eta \ge 0$. We show that the initial-value problems $ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ and $ u_t + \frac{1}{2} (u_x)^2 + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ are globally well-posed in $H^s(\mathbb{R})$, $s \ge 1$, $\eta>0$. We study the limiting behavior of the solutions of the first equation as $\eta$ tends to zero in $H^s(\mathbb{R})$ and $s \ge 2$. Moreover, we prove a unique continuation theorem for the first equation in $\mathcal F_{3,3}(\mathbb{R})=H^3(\mathbb{R}) \cap L^2_3(\mathbb{R})$, $\eta>0$.

Article information

Source
Differential Integral Equations, Volume 16, Number 10 (2003), 1249-1280.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060547

Mathematical Reviews number (MathSciNet)
MR2014809

Zentralblatt MATH identifier
1076.35108

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35G25: Initial value problems for nonlinear higher-order equations

Citation

Samaniego, Borys Alvarez. The Cauchy problem for a nonlocal perturbation of the KdV equation. Differential Integral Equations 16 (2003), no. 10, 1249--1280. https://projecteuclid.org/euclid.die/1356060547


Export citation