1999 Fiscal Year Final Research Report Summary
Singularly perturbed solutions of reaction-diffusion systems and concentration phenomena
| Project/Area Number |
09440046
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | TOHOKU UNIVERSITY |
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Principal Investigator |
TAKAGI Izumi Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (40154744)
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| Co-Investigator(Kenkyū-buntansha) |
NAGASAWA Takeyuki Graduate School of Science, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (70202223)
TSUTSUMI Yoshio Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (10180027)
NISHIURA Yasumasa Research Institute for Electronic Science, Hokkaido University, Professor, 電子科学研究所, 教授 (00131277)
TACHIZAWA Kazuya Graduate School of Science, Tohoku University, Lecturer, 大学院・理学研究科, 講師 (80227090)
IIDA Masato Faculty of Education, Iwate University, Lecturer, 教育学部, 講師 (00242264)
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| Project Period (FY) |
1997 – 1999
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| Keywords | reaction-diffusion system / singular perturbation / spike-layer solutions / transition layer / stability |
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| Research Abstract |
1. Takagi considered the construction and stability of stationary solutions of a reaction-diffusion system of activator-inhibitor type. With the cooperation of Wei-Ming Ni and Eiji Yanagida, he proved the following in the case of one dimensional domains : (i) The existence of stationary solutions concentrating at the boundary point when the activator diffuses slowly and the inhibitor diffuses very fast. (ii) If the relaxation parameter of the inhibitor reaction is small then these solutions are stable ; while they are unstable if the relaxation parameter is sufficiently large. (iii) A one-parameter family of periodic solutions concentrating around the boundary point bifurcates from the stationary solution.
Moreover, these results are generalized to higher dimensional domains in the case where the diffusion rate of the inhibitor is infinite.
2. Nishiura and Iida studied the behavior of solutions to the initial-boundary value problem for reaction-diffusion systems which generate sharp transition layers. Nishiura established a theory to explain the mechanism of self-replicating patterns. Iida constructed a reaction-diffusion system whose singular limit reduces to the classical Stefan problem.
3. Tsutsumi, Tachizawa and Nakano studied Schroedinger equations by applying techniques in real analysis. They obtained new results on the well-posedness of the initial value problem, and on the asymptotic distribution of eigenvalues.
4. Masuda and Nagasawa considered mainly the behavior of solutions to nonlinear diffusion equations. Masuda proved the maximum principle for weak solutions. Nagasawa refined the energy inequality for weak solutions to the Navier-Stokes equations.
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