Periodic traveling waves of a mean curvature flow in heterogeneous media

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February 2009, 25(1): 231-249. doi: 10.3934/dcds.2009.25.231

Periodic traveling waves of a mean curvature flow in heterogeneous media

1.	Department of Mathematics, Tongji University, Shanghai 200092

Received February 2007 Published June 2009

Abstract
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We consider a curvature flow in heterogeneous media in the plane: $ V= a(x,y) \kappa + b$, where for a plane curve, $V$ denotes its normal velocity, $\kappa$ denotes its curvature, $b$ is a constant and $a(x,y)$ is a positive function, periodic in $y$. We study periodic traveling waves which travel in $y$-direction with given average speed $c \geq 0$. Four different types of traveling waves are given, whose profiles are straight lines, ''V"-like curves, cup-like curves and cap-like curves, respectively. We also show that, as $(b,c)\rightarrow (0,0)$, the profiles of the traveling waves converge to straight lines. These results are connected with spatially heterogeneous version of Bernshteĭn's Problem and De Giorgi's Conjecture, which are proposed at last.

Keywords: curved front, heterogeneous media., Mean curvature flow, periodic traveling wave.

Mathematics Subject Classification: Primary: 35B27, 53C44; Secondary: 35B4.

Citation: Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

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