  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.
