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

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• 作者：Koichi Anadaup>aup>up> ; up>up>anada-koichi@waseda.jp" class="auth_mail" title="E-mail the corresponding authorup> ; Tetsuya Ishiwataup>bup>up> ; up>up>tisiwata@shibaura-it.ac.jp" class="auth_mail" title="E-mail the corresponding authorup>
• 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, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302960&_mathId=si1.gif&_user=111111111&_pii=S0022039616302960&_rdoc=1&_issn=00220396&md5=c65b3cc4f1bb1367ca5c3d044f06c816" title="Click to view the MathML source">kub>tub>=kup>2up>(kub>θθub>+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 urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302960&_mathId=si2.gif&_user=111111111&_pii=S0022039616302960&_rdoc=1&_issn=00220396&md5=5399adf038cac160c0ff217a476f5350">urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616302960-si2.gif">. In this paper, it is proved that urce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302960&_mathId=si3.gif&_user=111111111&_pii=S0022039616302960&_rdoc=1&_issn=00220396&md5=ba3f171b8bb109ad05f571c469fe358b">urce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616302960-si3.gif"> as ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302960&_mathId=si4.gif&_user=111111111&_pii=S0022039616302960&_rdoc=1&_issn=00220396&md5=ef86cd7dcf98d42f17fcf7add8222f6b" title="Click to view the MathML source">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.