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

Koichi Anada; Tetsuya Ishiwata

doi:10.1016/j.jde.2016.09.023

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

Koichi Anada^*, Tetsuya Ishiwata

^*Corresponding author for this work

Waseda University Senior High School

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

We consider initial-boundary value problems for a quasi linear parabolic equation, k_t=k²(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)⁻¹log⁡log⁡(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	https://doi.org/10.1016/j.jde.2016.09.023
Publication status	Published - 2017 Jan 5

Keywords

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

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2016.09.023

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@article{617587b02c3e454585feb8348090782d,

title = "Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation",

abstract = "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 supθ⁡k(θ,t)≈(T−t)−1log⁡log⁡(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. Vel{\'a}zquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate.",

keywords = "Curve shortening flows, Quasi-linear parabolic equations, Type II blow-up",

author = "Koichi Anada and Tetsuya Ishiwata",

note = "Funding Information: This work was finished while the first author was a visiting researcher at Shibaura Institute of Technology. This work was supported by JSPS KAKENHI Grant Number 15H03632 and 15K13461 . Appendix A Publisher Copyright: {\textcopyright} 2016 Elsevier Inc.",

year = "2017",

month = jan,

day = "5",

doi = "10.1016/j.jde.2016.09.023",

language = "English",

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journal = "Journal of Differential Equations",

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

AU - Anada, Koichi

AU - Ishiwata, Tetsuya

N1 - Funding Information: This work was finished while the first author was a visiting researcher at Shibaura Institute of Technology. This work was supported by JSPS KAKENHI Grant Number 15H03632 and 15K13461 . Appendix A Publisher Copyright: © 2016 Elsevier Inc.

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 supθ⁡k(θ,t)≈(T−t)−1log⁡log⁡(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 supθ⁡k(θ,t)≈(T−t)−1log⁡log⁡(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

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DO - 10.1016/j.jde.2016.09.023

M3 - Article

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VL - 262

SP - 181

EP - 271

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 1

ER -

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

Abstract

Keywords

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