Non-uniform dependence on initial data for the generalized Camassa-Holm-Novikov equation in Besov space

TL;DR

This paper proves that the solution map for the generalized Camassa-Holm-Novikov equation is not uniformly continuous in certain Besov spaces, extending and improving previous results on the equation's dependence on initial data.

Contribution

It establishes non-uniform dependence of solutions on initial data for a broad class of Besov spaces, including critical spaces, for the generalized Camassa-Holm-Novikov equation.

Findings

Solution map is not uniformly continuous in specified Besov spaces.

Results cover and improve previous work by Li et al.

Applicable to high-order nonlinear generalized Camassa-Holm-Novikov equation.

Abstract

Considered in this paper is the generalized Camassa-Holm-Novikov equation with high order nonlinearity, which unifies the Camassa-Holm and Novikov equations as special cases. We show that the solution map of generalized Camassa-Holm-Novikov equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$ as well as in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R}).$ Our result covers and improves the previous work given by Li et al. \cite{Li 2020, 1Li 2020, Li 2021}(J. Differ. Equ. 269 (2020) 8686-8700; J. Math. Fluid Mech. 22 (2020) 4:50; J. Math. Fluid Mech., (2021) 23:36).