Tohoku Mathematical Journal
2007
September
SECOND SERIES VOL. 59, NO. 3

Tohoku Math. J.
59 (2007), 365-377

Title PERIODIC TRAVELLING WAVE SOLUTIONS OF A CURVATURE FLOW EQUATION IN THE PLANE

Author Bendong Lou

(Received December 20, 2005, revised February 8, 2007)
Abstract. In the plane, we consider a curvature flow equation in heterogeneous media with periodic horizontal striations, the periodicity in space is expressed by periodic (in vertical direction) coefficients in the equation. We prove the existence and uniqueness of a curve which travels upward periodically with an average speed. At each time, the graph of the curve is a periodic undulating line at a finite distance from a straight line with a given inclination angle. We also show that the average speed depends on the inclination angle monotonously. Moreover, for homogenization problem as the spatial period tends to zero, we estimate the average speed by the inclination angle and some means of the periodic coefficients.

2000 Mathematics Subject Classification. Primary 35K55; Secondary 35B27, 35B10.
Key words and phrases. Periodic travelling wave solutions, curvature flow equation, homogenization problem.

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