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

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 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

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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 -