Linear damping around inhomogeneous stationary states of the Vlasov-HMF model

We will consider the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion. Such stationary states are closely related to the dynamics of the pendulum system. We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear damping effect with an algebraic rate of damping.