Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space

2023/02/06 by Deng, Wei, Li, Min, Wu, Xing +1
#35Q53 #37K10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.00887

Abstract

In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space B¹_∞,1(\R). It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa-Holm equation is ill-posed in B¹_∞,1(\R), here we turn our attention to a higher-order nonlinear generalization of Camassa-Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space B¹_∞,1(\R) which leads to ill-posedness.

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