Asymptotic stability of peakons for the Novikov equation

The Novikov equation is a Camassa-Holm type equation with cubic nonlinearity. This paper aims to prove the asymptotic stability of peakons solutions under $H^1(\mathbb{R})$-perturbations satisfying that their associated momentum density defines a non-negative Radon measure. Motivated by Molinet's work, we shall first prove a Liouville property for $H^1(\mathbb{R})$ global solutions belonging to a certain class of almost localized functions. More precisely, we show that such solutions have to be a peakon. The main difficulty in our analysis in comparison to the Camassa-Holm case comes from the fact that the momentum is not conserved and may be unbounded along the trajectory. Also, to prove the Liouville property, we used a new Lyapunov functional not related to the (not conserved) momentum of the equation.