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 H¹(R) to one defined on weaker functions on L²(R), we prove spectral and linear instability on of peakons in the b-Novikov equations for any b. We also consider the stability on and show that the peakons are spectrally or linearly stable only in the case $b=3$.