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: S1× [0,T) → R2 which satisfies the following evolution equation
∂2 F/ ∂ t 2 (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 ＝(∂2F/ ∂s ∂t, ∂F/ ∂t ) T,
in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampère equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax ) and when t goes to Tmax, either the solution converges to a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R1,1.
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