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

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 $W^{1,\mathrm{\infty }}$-perturbations. Moreover, we show that small initial $W^{1,\mathrm{\infty }}$-perturbations of the Novikov peakons can lead to the finite time blow-up of the corresponding solutions.