# American Institute of Mathematical Sciences

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

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