  It is known from the previous works that the peakon solutions of the Novikov
equation are orbitally and asymptotically stable in $H^1$. We prove, via the
method of characteristics, that these peakon solutions are unstable under
$W^{1,\infty}$-perturbations. Moreover, we show that small initial
$W^{1,\infty}$-perturbations of the Novikov peakons can lead to the finite time
blow-up of the corresponding solutions.
