Ill-Posedness of the Novikov Equation in the Critical Besov Space \(B^{1}_{\infty ,1}(\mathbb {R})\)

It is shown that both the Camassa–Holm and Novikov equations are ill-posed in \(B_{p,r}^{1+1/p}(\mathbb {R})\) $B_{p,r}^{1+1/p}(\mathbb{R})$ with \((p,r)\in [1,\infty ]\times (1,\infty ]\) $(p,r)\in [1,\infty ]\times (1,\infty ]$ in Guo et al. (J Differ Equ 266:1698–1707, 2019) and well-posed in \(B_{p,1}^{1+1/p}(\mathbb {R})\) $B_{p,1}^{1+1/p}(\mathbb{R})$ with \(p\in [1,\infty )\) $p\in [1,\infty )$ in Ye et al. (J Differ Equ 367: 729–748, 2023). Recently, the ill-posedness for the Camassa–Holm equation in \(B^{1}_{\infty ,1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ has been proved in Guo et al. (J Differ Equ 327: 127–144, 2022). In this paper, we shall solve the only left an endpoint case \(r=1\) $r=1$ for the Novikov equation. More precisely, we prove the ill-posedness for the Novikov equation in \(B^{1}_{\infty ,1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ by exhibiting the norm inflation phenomena.