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.