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Hyperbolic mean curvature flow with a forcing term: evolution of plane curves. (English) Zbl 1285.53057

Summary: In this paper we study the evolution of closed strictly convex plane curves moving by the hyperbolic mean curvature flow with a forcing term. It is shown that the flow admits a unique short-time smooth solution and the convexity of the curves is preserved during the evolution. When the forcing term is a negative constant, we prove the curves either converge to a point or a \(C^0\) curve. For a positive constant forcing term, the flow has a unique smooth solution in any finite time and expands to infinity as \(t\) tends to infinity if the initial curvature is smaller than \(M\), the flow will blow up in a finite time if the initial curvature is larger than \(M\).

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J45 Hyperbolic equations on manifolds
58J47 Propagation of singularities; initial value problems on manifolds
35L70 Second-order nonlinear hyperbolic equations
35L45 Initial value problems for first-order hyperbolic systems
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