Abstract:To study the dynamics of an anisotropic curvature flow with external driving force depending only on the normal vector, we focus on traveling waves composed of Jordan curves in . Here we call them compact traveling waves. The objective of this study is to investigate thoroughly the condition of the driving force for the existence of compact traveling waves to the anisotropic curvature flow. It is shown that all traveling waves are strictly convex and unstable, and that a compact traveling wave is unique, if they exist. To determine the existence of compact traveling waves, three cases are considered: if the driving force is positive, there exists a compact traveling wave; if it is negative, there is no traveling wave; if it is sign-changing, a positive answer is obtained under the assumption called ``admissible condition''. We also obtain a necessary and sufficient condition for the existence of axisymmetric compact traveling waves. Lastly, we make reference to the inverse problem and non-convex compact traveling waves.

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中文翻译：

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各向异性曲率的紧凑行波随驱动力流动

摘要：为了研究仅依赖于法向矢量的，具有外部驱动力的各向异性曲率流的动力学，我们集中于由乔丹曲线构成的行波。在这里，我们称它们为紧凑的行波。这项研究的目的是彻底研究致密向各向异性曲率流传播的行波的驱动力条件。结果表明，所有的行波都是严格凸且不稳定的，并且紧凑的行波（如果存在）是唯一的。为了确定致密行波的存在，考虑三种情况：如果驱动力为正，则存在致密行波；如果驱动力为正，则存在致密行波。如果为负，则没有行波；如果它正在改变符号，则在称为`` 可接受的条件”。我们还为存在轴对称紧凑行波获得了充要条件。最后，我们参考了反问题和非凸紧凑行波。

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