Spectral instability of peakons for the $b$-family of Novikov equations

In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter $b$. This family is called the ``$b$-Novikov'' since it reduces to the integrable Novikov equation in the case $b=3$. By extending the corresponding linearized operator defined on functions in $H^1(\mathbb{R})$ to one defined on weaker functions on $L^2(\mathbb{R})$, we prove spectral and linear instability on $L^2(\mathbb{R})$ of peakons in the $b$-Novikov equations for any $b$. We also consider the stability on $H^1(\mathbb{R})$ and show that the peakons are spectrally or linearly stable only in the case $b=3$.