### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

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

**Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation.** / Anada, Koichi; Ishiwata, Tetsuya.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 262, no. 1, pp. 181-271. https://doi.org/10.1016/j.jde.2016.09.023

}

TY - JOUR

T1 - Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation

AU - Anada, Koichi

AU - Ishiwata, Tetsuya

PY - 2017/1/5

Y1 - 2017/1/5

N2 - We consider initial-boundary value problems for a quasi linear parabolic equation, kt=k2(kθθ+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 supθk(θ,t)≈(T−t)−1loglog(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.

AB - We consider initial-boundary value problems for a quasi linear parabolic equation, kt=k2(kθθ+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 supθk(θ,t)≈(T−t)−1loglog(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.

KW - Curve shortening flows

KW - Quasi-linear parabolic equations

KW - Type II blow-up

UR - http://www.scopus.com/inward/record.url?scp=84994411688&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84994411688&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2016.09.023

DO - 10.1016/j.jde.2016.09.023

M3 - Article

AN - SCOPUS:84994411688

VL - 262

SP - 181

EP - 271

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -