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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.

具有幂非线性的半线性热方程在Sobolev临界指数前后解的性质发生了显著变化。在亚临界情况下对它们进行了详细的研究，但在超临界情况下只有很少的结果。不完全爆破、整体解长大和II型爆破是超临界情况下的典型现象。不完全爆破解的存在是已知的，但爆破后的行为提醒了开放。得到了经典意义上两次爆破的弱解，并将其推广到一般的多次爆破。这些解在每次爆破时间在同一点（原点）爆破，两次连续爆破时间之差足够大。在给定的爆破时间下，研究了在不同爆破时间在不同地点爆破的解的存在性。Galaktionov-Vazquez和我，有一个峰值解，当时间趋于无穷时收敛于0。我们给出了无穷时刻各种行为的峰值解。应用基于无限维动力系统的方法，构造了具有精确生长速率的生长解。并给出了奇异稳态下解的最优生长速率。委拉斯开兹证明了存在一种溶液，它经历了II型爆炸。然而，他们的证明非常冗长和复杂。我们得到了一个II型爆破解作为整体解和爆破解之间的分离。我们的证明比上一个简单。