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

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• 作者：Koichi Anada^a ; ^{anada-koichi@waseda.jp" class="auth_mail" title="E-mail the corresponding author} ; Tetsuya Ishiwata^b ; ^{tisiwata@shibaura-it.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Type II blow-up ; Quasi-linear parabolic equations ; Curve shortening flows
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 January 2017
• 年：2017
• 卷：262
• 期：1
• 页码：181-271
• 全文大小：869 K

文摘

We consider initial-boundary value problems for a quasi linear parabolic equation, cc4f1bb1367ca5c3d044f06c816" title="Click to view the MathML source">k_t=k²(k_θθ+k)$k_t=k^2(k_{\theta \theta }+k)$, 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 $\sqrt{{(T-t)}^{-1}}$. In this paper, it is proved that ${\sup }_{\theta }k(\theta ,t)\approx \sqrt{{(T-t)}^{-1}\log \log {(T-t)}^{-1}}$ as t↗T$t\nearrow T$ 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.