The \(\mu \)-Camassa–Holm equation and the modified \(\mu \)-Camassa–Holm equation, as the nonlocal counterparts of the Camassa–Holm and modified Camassa–Holm equations, are two integrable models. Local well-posedness of the Cauchy problems for the two equations has been established in the space \(C([0,T), H^s(\mathbb {S}))\bigcap C^1([0,T), H^{s-1}(\mathbb {S}))\), respectively, when \(s>3/2\) and \(s>5/2\). This paper mainly studies ill-posedness of these two equations for \(s<3/2\). First, we show that the modified \(\mu \)-Camassa–Holm equation is ill-posed due to failure of uniqueness, which is carried out by constructing a 2-peakon solution with an asymmetric peakon-antipeakon initial profile and utilizing the time-reversibility of the equation. Second, by estimating the \(H^s\)-norm of peakon-antipeakon solutions as time tends to the lifespan, we investigate ill-posedness of the Cauchy problem for the \(\mu \)-CH equation, respectively for \(1<s<3/2,\) \(s=1\) and \(s<1\).

\ (\mu \) -Camassa–Holm 方程和修正的\(\mu \) -Camassa–Holm 方程作为 Camassa–Holm 方程和修正的 Camassa–Holm 方程的非局部对应项，是两个可积模型。两个方程的柯西问题的局部适定性已在空间\(C([0,T), H^s(\mathbb {S}))\bigcap C^1([0,T) 中建立， H^{s-1}(\mathbb {S}))\)，分别当\(s>3/2\)和\(s>5/2\)时。本文主要研究这两个方程对于\(s<3/2\)的不适定性。首先，我们证明修改后的\(\mu \)-Camassa-Holm方程由于唯一性失败而不适定，这是通过构造具有不对称peakon-antipeakon初始轮廓的2-peakon解并利用方程的时间可逆性来实现的。其次，随着时间趋于寿命，通过估计peakon-antipeakon解的\(H^s\) -范数，我们研究了\(\mu \) -CH方程的柯西问题的不适定性，分别为\ (1<s<3/2,\) \(s=1\)和\(s<1\)。