Hyperbolic mean curvature flow: Evolution of plane curves

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves F : S × [0, T ) → R which satisfies the following evolution equation ∂F ∂t (u, t) = k(u, t) ~ N(u, t)− ▽ρ(u, t), ∀ (u, t) ∈ S1 × [0, T ) with the initial data F (u, 0) = F0(u) and ∂F ∂t (u, 0) = f(u) ~ N0, where k is the mean curvature and ~ N is the unit inner normal vector of the plane curve F (u, t), f(u) and ~ N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0 respectively, and ▽ρ is given by ▽ρ , fi ∂F ∂s∂t , ∂F ∂t fl