### Abstract

We consider the expansion of a convex closed plane curve C_{0} along its outward normal direction with speed G(1/k), where k is the curvature and (Formula presented.) is a strictly increasing function. We show that if (Formula presented.), then the isoperimetric deficit (Formula presented.) of the flow converges to zero. On the other hand, if (Formula presented.), then for any number d ≥ 0 and (Formula presented.), one can choose an initial curve C_{0} so that its isoperimetric deficit D(t) satisfies (Formula presented.) for all t ∈[0, ∞). Hence, without rescaling, the expanding curve C_{t} will not become circular. It is close to some expanding curve P_{t}, where each P_{t} is parallel to P_{0}. The asymptotic speed of P_{t} is given by the constant λ.

Original language | English |
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Pages (from-to) | 779-794 |

Number of pages | 16 |

Journal | Journal of Evolution Equations |

Volume | 14 |

Issue number | 4-5 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

### Cite this

*Journal of Evolution Equations*,

*14*(4-5), 779-794. https://doi.org/10.1007/s00028-014-0238-2