Life-span of Hyperbolic Mean Curvature Flow in Minkowski Space $\mathbf{R}^{1,1}$

This paper considers with the evolution of the spacelike curve in Minkowski space \(\mathbf{R}^{1,1}\) under hyperbolic mean curvature flow. Firstly, the quasilinear hyperbolic system under hyperbolic mean curve flow motion and its basic properties are derived. Then, based on the assumptions of initial data and some constraints, the life-span estimation of the quasilinear hyperbolic system are discussed. Finally, it is proved that the \(C^2\) smooth solution of hyperbolic mean curvature flow will certainly blow-up in limited time and the life-span estimation are given.