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

Koichi Anada, Tetsuya Ishiwata

Research output: Contribution to journalArticle

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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. Velázquez. In addition, we prove some of numerical conjectures by Watterson which are keys to provide the blow-up rate.

LanguageEnglish
Pages181-271
Number of pages91
JournalJournal of Differential Equations
Volume262
Issue number1
DOIs
StatePublished - 2017 Jan 5

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

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

In: Journal of Differential Equations, Vol. 262, No. 1, 05.01.2017, p. 181-271.

Research output: Contribution to journalArticle

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