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2003 The Cauchy problem for a nonlocal perturbation of the KdV equation
Borys Alvarez Samaniego
Differential Integral Equations 16(10): 1249-1280 (2003).

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 $\mathcal F_{3,3}(\mathbb{R})=H^3(\mathbb{R}) \cap L^2_3(\mathbb{R})$, $\eta>0$.

Citation

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Borys Alvarez Samaniego. "The Cauchy problem for a nonlocal perturbation of the KdV equation." Differential Integral Equations 16 (10) 1249 - 1280, 2003.

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1076.35108
MathSciNet: MR2014809

Subjects:
Primary: 35Q53
Secondary: 35G25

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.16 • No. 10 • 2003
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