A dissipative hyperbolic affine curve flow

A new hyperbolic affine geometric flow with dissipative term is proposed. The equations satisfied by support functions and the graph of the curve under this dissipative flow give rise to fully nonlinear hyperbolic equations, we obtain the existence for local solutions of this flow by reducing the flow to a first-order system. The equations for both perimeter and area of closed curves under the flow are also obtained. Based on this, we show that for a closed curve, the solution of this flow converges to a point in finite time. Furthermore, by a method of LeFloch-Smocyk for studying the hyperbolic mean curvature flow, global existence of the solution is established.