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

Mitsunori Nara, Masaharu Taniguchi

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)203-220
Number of pages18
JournalDiscrete and Continuous Dynamical Systems
Volume14
Issue number1
DOIs
Publication statusPublished - Jan 1 2006
Externally publishedYes

Keywords

  • Curvature flow
  • Stability
  • Traveling waves

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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