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

详细信息    查看全文

• 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, class="mathmlsrc">class="formulatext 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">k_t=k²(k_θθ+k)class="mathContainer hidden">class="mathCode">$k_t=k^2(k_{\theta \theta }+k)$, 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 class="mathmlsrc">title="View the MathML source" 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">class="imgLazyJSB inlineImage" height="21" width="81" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616302960-si2.gif">class="mathContainer hidden">class="mathCode">$\sqrt{{(T-t)}^{-1}}$. In this paper, it is proved that class="mathmlsrc">title="View the MathML source" 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">class="imgLazyJSB inlineImage" height="21" width="283" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022039616302960-si3.gif">class="mathContainer hidden">class="mathCode">${\sup }_{\theta }k(\theta ,t)\approx \sqrt{{(T-t)}^{-1}\log \log {(T-t)}^{-1}}$ as class="mathmlsrc">class="formulatext 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↗Tclass="mathContainer hidden">class="mathCode">$t\nearrow T$ 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.