On the singularity of solutions for nonlinear parabolic equation

非线性抛物方程解的奇异性

基本信息

批准号：
17540189
负责人：
MIZOGUCHI Noriko
金额：
$ 2.24万
依托单位：
Tokyo Gakugei University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540189/
关键词：
semilinear heat equation incomplete blowup supercritical exponent multiple blowup type II blowup blowup rate

项目摘要

The behavior of solutions to a semilinear heat equation with power nonlinearity dramatically changes before and after the Sobolev critical exponent. They are investigated in detail in the subcritical case, but there are only a few results in the supercritical case. Incomplete blowup, growup of a global solution and type II blowup are typical phenomena in the superciritical case.The existence of a incomplete blowup solution is known, but the behavior after blowup has reminded open. We obtained a weak solution which blowup twice in the classical sense, and extended it to general multiple blowup. These solutions blow up at the same point (at the origin) at each blowup time and the difference between two successive blowup times is sufficiently large. We studied the existence of a solution which blows up at different places at different blowup times for given blowup times.By Galaktionov-Vazquez and myself, there were peaking solution which converges to 0 as time tens to infinity. We gave peaking solutions with various behaviors at time infinity.We constructed growup solutions with exact growup rate applying a method based on infinite-dimensional dynamical system. It also gave the optimal growup rate of solutions below the singular steady state.Herrero-Velazquez showed that there exists a solution which undergoes type II blowup. However their proof is very long and complicated. We got a type II blowup solution as a separatirix between global solutions and blowup solutions. Our proof is simpler than the previous one.

在Sobolev临界指数之前和之后，具有功率非线性的半线性热方程解决方案的行为发生了巨大变化。在亚临界情况下，对它们进行了详细研究，但是在超批评中只有少数结果。在超严重情况下，不完整的爆炸，全球解决方案的逐渐成长和II型爆炸是典型的现象。已知不完整的爆炸解决方案的存在，但是爆炸后的行为已提醒。我们获得了一个薄弱的解决方案，该解决方案在经典意义上两次爆炸，并将其扩展到一般的多重爆炸。这些解决方案在每个爆炸时间在同一点（原点）爆炸，并且两个连续的爆炸时间之间的差异足够大。我们研究了一种解决方案的存在，该解决方案在不同的爆炸时间在不同的爆炸时间炸毁了。通过Galaktionov-Vazquez和我本人，有峰值解决方案，这些解决方案趋于0，因为时间数十个无限。我们在时间无穷大时给出了具有各种行为的峰值解决方案。我们构建了基于无限维动力学系统的方法的精确增长速率。它还给出了在单数稳态以下的最佳溶液增长速率。Herrero-Velazquez表明存在一种经历II型爆炸的溶液。但是，他们的证明非常长且复杂。我们在全局解决方案和爆炸解决方案之间获得了II型爆炸解决方案。我们的证明比上一个更简单。