Dissipative hyperbolic mean curvature flow for closed convex plane curves

Abstract

This paper considers one dimensional dissipative hyperbolic mean curvature flow. To analysis the asymptotic behavior of this dissipative hyperbolic curvature flow, we investigate the evolution of a family of circles. Particularly, we give some propositions and prove the main result. If the minimum of initial velocity is nonnegative, the flow will converge either to a point or a limit curve which has the discontinuous curvature in finite time. If the maximum of initial velocity is positive, the flow will expand firstly and then converge either to a point or a limit curve which has the discontinuous curvature in finite time.