2006 Fiscal Year Final Research Report Summary
Asymptotic behavior of solutions for some diffusive equations and its applications
| Project/Area Number |
15540202
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Global analysis
|
|---|
| Research Institution | Tohoku University (2004-2006)
Nagoya University (2003) |
|---|
Principal Investigator |
ISHIGE Kazuhiro Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90272020)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YANAGIDA Eiji Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
KOZONO Hideo Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00195728)
OGAWA Takayoshi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20224107)
HATTORI Tetsuya Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10180902)
|
|---|
| Project Period (FY) |
2003 – 2006
|
| Keywords | heat equation / blow-up problem / movement of hot spots / exterior domain / eigenfunction |
|---|
| Research Abstract |
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.
|
