Sharp well-posedness for the Chen-Lee equation

Ricardo A.  Pastrán; Oscar G.  Riaño

doi:10.3934/cpaa.2016033

Article Contents

2016, Volume 15, Issue 6: 2179-2202. Doi: 10.3934/cpaa.2016033

This issue Previous Article Sign-changing solutions for fourth order elliptic equations with Kirchhoff-type Next Article Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

Sharp well-posedness for the Chen-Lee equation

Ricardo A. Pastrán^1, and
Oscar G. Riaño^2,

1.
Departamento de Matemáticas, Universidad Nacional de Colombia, AK 30 45-03, 111321, Bogotá
2.
Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ

Received: November 30, 2015

Revised: April 30, 2016

Published: October 2016

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Abstract

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(\mathbb{R})$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.

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Mathematics Subject Classification: Primary: 35A01, 35K55, 35Q53; Secondary: 76B15, 76B03.

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