Differential and Integral Equations

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

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

https://projecteuclid.org/euclid.die/1356012251

Mathematical Reviews number (MathSciNet)
MR3013404

Zentralblatt MATH identifier
1274.35342

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