American Institute of Mathematical Sciences

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

Periodic traveling waves of a mean curvature flow in heterogeneous media

Bendong Lou 1,

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.
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 and Continuous Dynamical Systems, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231
[1]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[2]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[3]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[4]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016
[5]

Fathi Dkhil, Angela Stevens. Traveling wave speeds in rapidly oscillating media. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 89-108. doi: 10.3934/dcds.2009.25.89
[6]

Tobias H. Colding and Bruce Kleiner. Singularity structure in mean curvature flow of mean-convex sets. Electronic Research Announcements, 2003, 9: 121-124.

[7]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure and Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139
[8]

Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049
[9]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[10]

Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
[11]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[12]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
[13]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure and Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307
[14]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463
[15]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[16]

Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1
[17]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[18]

Shi-Liang Wu, Cheng-Hsiung Hsu. Propagation of monostable traveling fronts in discrete periodic media with delay. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2987-3022. doi: 10.3934/dcds.2018128
[19]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[20]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (98)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy