March  2006, 16(1): 137-156. doi: 10.3934/dcds.2006.16.137

Convergence to V-shaped fronts 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

2. 

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

Received  August 2005 Revised  February 2006 Published  June 2006

This paper is concerned with the long time behavior for evolution of a curve governed by a curvature flow with constant driving force in the two-dimensional space. This problem has two types of traveling waves: traveling lines and V-shaped fronts, except for stationary circles. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the $x$-axis. In this paper, we consider the uniform convergence of curves to the V-shaped fronts. Convergence results for a class of spatially non-decaying initial perturbations are established. Our results hold true with no assumptions on the smallness of given perturbations.
Citation: Mitsunori Nara, Masaharu Taniguchi. Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 137-156. doi: 10.3934/dcds.2006.16.137
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