The Cauchy problem of dissipative hyperbolic mean curvature flow

Abstract

In this paper, the motion of strictly convex closed plane curves under dissipative hyperbolic mean curvature flow is studied. The hyperbolic Monge–Amp re equation is derived by using the support function. The short‐time existence of the flow is proved, and some evolution equations are derived. Furthermore, according to different initial velocities, we discuss the expansion and contraction of the dissipative hyperbolic curvature flow; that is, if the initial velocity , the flow will converge to the limit curve at a finite time; if the initial velocity , the flow will converge to a point ; if the initial velocity , the flow will contract to a limit curve as ; if the initial velocity , the flow will expand to a limit curve as .