January  2006, 14(1): 203-220. doi: 10.3934/dcds.2006.14.203

Stability of a traveling wave in curvature flows for spatially non-decaying initial perturbations

1. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama 2-12-1-W8-38, Meguro-ku, Tokyo 152-8552, Japan

2. 

Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552

Received  November 2004 Revised  March 2005 Published  October 2005

This paper is concerned with the long time behavior for the evolution of a curve governed by the curvature flow with constant driving force in two-dimensional space. Especially, the asymptotic stability of a traveling wave whose shape is a line is studied. We deal with moving curves represented by the entire graphs on the $x$-axis. By studying the Cauchy problem, the asymptotic stability of traveling waves with spatially decaying initial perturbations and the convergence rate are obtained. Moreover we establish the stability result where initial perturbations do not decay to zero but oscillate at infinity. In this case, we prove that one of the sufficient conditions for asymptotic stability is that a given perturbation is asymptotic to an almost periodic function in the sense of Bohr at infinity. Our results hold true with no assumptions on the smallness of given perturbations, and include the curve shortening flow problem as a special case.
Citation: Mitsunori Nara, Masaharu Taniguchi. Stability of a traveling wave in curvature flows for spatially non-decaying initial perturbations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 203-220. doi: 10.3934/dcds.2006.14.203
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