Differential and Integral Equations

Unique continuation principles for the Benjamin-Ono equation

Rafael José Iorio

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Abstract

Let $\sigma$ denote the Hilbert transform and $\mu\geq0$. We prove that if $u\in C\left( \left[ 0,T\right] ,H^{2}\left( \mathbb{R}\right) \cap L_{2}^{2}\left( \mathbb{R}\right) \right) $ is a solution of \[ \partial_{t}u+2\sigma\partial_{x}^{2}u+u\partial_{x}u=\mu\partial_{x}^{2}u \] such that there are $t_{0} <t_{1}$ $ <t_{2\text{ }}$satisfying $u\left( t_{j}\right) \in H^{4}\left( \mathbb{R}\right) \cap L_{4}^{2}\left( \mathbb{R}\right) $, $j=0,1,2$, then $u\left( t\right) =0$ for all $t\in\left[ 0,T\right] $.

Article information

Source
Differential Integral Equations, Volume 16, Number 11 (2003), 1281-1291.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060510

Mathematical Reviews number (MathSciNet)
MR2016683

Zentralblatt MATH identifier
1075.35552

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Iorio, Rafael José. Unique continuation principles for the Benjamin-Ono equation. Differential Integral Equations 16 (2003), no. 11, 1281--1291. https://projecteuclid.org/euclid.die/1356060510


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