  In this paper we investigate the one-dimensional hyperbolic mean curvature
flow for closed plane curves. We show that there exists a class of initial
velocities such that the solution of the corresponding initial value problem
exists only at a finite time interval $[0,T_{\max})$ and when $t$ goes to
$T_{\max}$, the solution converges to a point. We also discuss the close
relationship between the hyperbolic mean curvature flow and the equations for
the evolving relativistic string in the Minkowski space-time
$\mathbb{R}^{1,1}$.
