The condition on the stability of stationary lines in a curvature flow in the whole plane

Mitsunori Nara, Masaharu Taniguchi

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O (t- 1 / 2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.

Original languageEnglish
Pages (from-to)61-76
Number of pages16
JournalJournal of Differential Equations
Volume237
Issue number1
DOIs
Publication statusPublished - Jun 1 2007
Externally publishedYes

Fingerprint

Curvature Flow
Heat Equation
Curve
Line
Cauchy Problem
Moving Planes
Decay Estimates
Long-time Behavior
Infinity
Entire
Converge
Necessary Conditions
Derivative
Sufficient Conditions
Graph in graph theory
Derivatives
Hot Temperature

Keywords

  • Asymptotic behavior
  • Curvature flow
  • Heat equation

ASJC Scopus subject areas

  • Analysis

Cite this

The condition on the stability of stationary lines in a curvature flow in the whole plane. / Nara, Mitsunori; Taniguchi, Masaharu.

In: Journal of Differential Equations, Vol. 237, No. 1, 01.06.2007, p. 61-76.

Research output: Contribution to journalArticle

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