Differential and Integral Equations

Sharp well-posedness and ill-posedness of a higher-order modified Camassa--Holm equation

Yongsheng Li, Shiming Li, and Wei Yan

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

In this paper we consider the Cauchy problem for a higher-order modified Camassa--Holm equation. By using some dyadic bilinear estimates and the fixed-point theorem, we establish the local well-posedness of the higher-order modified Camassa--Holm equation for the small initial data in $H^{-n+\frac{5}{4}}({{\mathbf R}}),$ $n\geq 2,$ $ n\in {{\mathbf N}}$. We also prove that the Cauchy problem for the higher-order modified Camassa--Holm equation is ill-posed for the initial data in homogeneous Sobolev spaces $\dot{H}^{s}({{\mathbf R}})$ with $s < -n+\frac{5}{4},$ $n\in {{\mathbf N}},$ $ n\geq 2$. Our result partially answers the open problem which is proposed below in Theorem 1.2 by Erika A. Olson in the Journal of Differential Equations, 246 (2009), 4154--4172.

Article information

Source
Differential Integral Equations, Volume 25, Number 11/12 (2012), 1053-1074.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012251

Mathematical Reviews number (MathSciNet)
MR3013404

Zentralblatt MATH identifier
1274.35342

Subjects
Primary: 35R25: Improperly posed problems 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems

Citation

Yan, Wei; Li, Yongsheng; Li, Shiming. Sharp well-posedness and ill-posedness of a higher-order modified Camassa--Holm equation. Differential Integral Equations 25 (2012), no. 11/12, 1053--1074. https://projecteuclid.org/euclid.die/1356012251


Export citation