### Abstract

We consider initial-boundary value problems for a quasi linear parabolic equation, k_{t}=k^{2}(k_{θθ}+k), with zero Dirichlet boundary conditions and positive initial data. It has known that each of solutions blows up at a finite time with the rate faster than (T−t)^{−1}. In this paper, it is proved that sup_{θ}k(θ,t)≈(T−t)^{−1}loglog(T−t)^{−1} as t↗T under some assumptions. Our strategy is based on analysis for curve shortening flows that with self-crossing brought by S.B. Angenent and J.J.L. Velázquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate.

Original language | English |
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Pages (from-to) | 181-271 |

Number of pages | 91 |

Journal | Journal of Differential Equations |

Volume | 262 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 Jan 5 |

### Keywords

- Curve shortening flows
- Quasi-linear parabolic equations
- Type II blow-up

### ASJC Scopus subject areas

- Analysis

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## Cite this

Anada, K., & Ishiwata, T. (2017). Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation.

*Journal of Differential Equations*,*262*(1), 181-271. https://doi.org/10.1016/j.jde.2016.09.023