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.

具有功率非线性的半线性热方程的解的行为在 Sobolev 临界指数之前和之后发生显着变化。它们在亚临界情况下进行了详细研究，但在超临界情况下只有很少的结果。不完全爆炸、全局解的增长和II型爆炸是超临界情况下的典型现象。不完全爆炸解的存在是已知的，但爆炸后的行为已经提醒人们开放。我们得到了经典意义上的两次爆炸的弱解，并将其推广到一般的多次爆炸。这些溶液在每个爆炸时间在同一点（在原点）爆炸，并且两个连续爆炸时间之间的差异足够大。我们研究了一个解的存在性，该解在给定的爆炸时间内在不同的爆炸时间在不同的地方爆炸。Galaktionov-Vazquez 和我自己发现了峰值解，当时间为十到无穷大时，它会收敛到 0。我们给出了无穷时间时各种行为的峰值解。我们应用基于无限维动力系统的方法构建了具有精确增长速率的增长解。它还给出了低于奇异稳态的解的最佳增长速率。Herrero-Velazquez 证明存在一个经历 II 型爆炸的解。然而他们的证明非常长且复杂。我们得到了 II 型爆炸解决方案，作为全局解决方案和爆炸解决方案之间的分离。我们的证明比前一个简单。