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Asymptotic behavior of solutions for some diffusive equations and its applications

某些扩散方程解的渐近行为及其应用

基本信息

批准号：
15540202
负责人：
ISHIGE Kazuhiro
金额：
$ 2.24万
依托单位：
Tohoku University (2004-2006)Nagoya University (2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

We studied the location of the blow-up set for the solutions for a semilinear heat equation with large diffusion, under the homogeneous Neumann boundary condition, in a bounded smooth domain of the Euclidean space. This was a joint work with Professors Noriko Mizoguchi and Hiroki Yagisita. We proved that, if the diffusive coefficient is sufficiently large, for almost all initial data, the solution blows-up in a finite time only near the maximum points of the projection of the initial data onto the second Neumann eigenspace. This is the first result that explains the relation between the eigenfunctions and the location of the blow-up set.On the other hand, we studied the movement of the maximum points (hot spots) of the solutions of the heat equations. In particular, we considered the solution for the Cauchy-Neumann problem and the Cauchy-Dirichlet problem to the heat equation in the exterior domain of a ball. This exterior domain is very simple, but it is difficult to study the movement of hot spots. By using harmonic functions, we obtained some good asymptotic behavior of the hot spots as the time tends to infinity. After that, we studied the decay rate of derivatives of the solution and the movement of hot spots for the solution of the heat equation, with Professor Yoshitsugu Kabeya. By this study, we can understand the mechanism how to decide the decay rate of the derivatives of the solutions and the movement of hot spots.

在齐次Neumann边界条件下，研究了具有大扩散项的半线性热方程解在欧氏空间中有界光滑区域上爆破集的位置.这是与沟口纪子教授和八木下博树教授的联合工作。我们证明了，如果扩散系数足够大，对于几乎所有的初始数据，只有在第二Neumann特征空间上的初始数据的投影的极大值点附近的解在有限时间内爆破。这是第一个解释本征函数与爆破集位置之间关系的结果。另一方面，我们研究了热方程解的极大值点（热点）的移动。特别地，我们考虑了球外区域上热方程的Cauchy-Neumann问题和Cauchy-Dirichlet问题的解。这一外部域非常简单，但研究热点的运动是困难的。利用调和函数，我们得到了热点在时间趋于无穷大时的一些好的渐近性质。之后，我们与Kabeya Yoshitsugu教授一起研究了热方程解的导数衰减率和热点的移动。通过这一研究，我们可以了解决定解的导数衰减速率和热点移动的机理。