$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation

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