Periodic travelling wave solutions of a curvature flow equation in the plane.

*(English)*Zbl 1138.35035This paper deals with the study of periodic travelling wave solutions of a curvature flow equation in the plane. The main result of this paper establishes the existence and the uniqueness of such a solution, whose graphic is a periodic ondulating line which is in a finite distance from a straight line with a prescribed inclination \(\alpha\), so that the propagation is just like that in oblique disposed striations. Two particular cases have a particular interest in this analysis. First, if \(\alpha =0\), then the periodic travelling wave solution is a horizontal straight line which travels in the \(y\)-direction with average speed \(c_0\). Next, in the case \(\alpha =\pi /2\), then there exists not-periodic travelling wave solution which travels in the -\(x\)-direction with a speed depending on the arithmetic means of two well-defined quantities.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

##### MSC:

35K55 | Nonlinear parabolic equations |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

35B10 | Periodic solutions to PDEs |

##### Keywords:

periodic travelling wave solutions; curvature flow equation; homogenization problem; existence; uniqueness; periodic ondulating line##### References:

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