### 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 language | English |
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Pages (from-to) | 61-76 |

Number of pages | 16 |

Journal | Journal of Differential Equations |

Volume | 237 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 1 2007 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 237, no. 1, pp. 61-76. https://doi.org/10.1016/j.jde.2007.02.012

}

TY - JOUR

T1 - The condition on the stability of stationary lines in a curvature flow in the whole plane

AU - Nara, Mitsunori

AU - Taniguchi, Masaharu

PY - 2007/6/1

Y1 - 2007/6/1

N2 - 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.

AB - 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.

KW - Asymptotic behavior

KW - Curvature flow

KW - Heat equation

UR - http://www.scopus.com/inward/record.url?scp=34247323978&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247323978&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2007.02.012

DO - 10.1016/j.jde.2007.02.012

M3 - Article

AN - SCOPUS:34247323978

VL - 237

SP - 61

EP - 76

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -