2018 Fiscal Year Final Research Report
Study on the profiles and dynamics of solutions to reaction-diffusion systems through the asymptotic analysis
| Project/Area Number |
15K04963
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | University of Miyazaki |
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Principal Investigator |
Iida Masato 宮崎大学, 工学部, 教授 (00242264)
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| Co-Investigator(Kenkyū-buntansha) |
二宮 広和 明治大学, 総合数理学部, 専任教授 (90251610)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 非線形解析 / 反応拡散系 / 漸近解 |
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| Outline of Final Research Achievements |
Reaction-diffusion systems are mathematical models describing the dynamics of the population densities of particles which appear, disappear and interact each other. Numerical simulations suggest that various dynamic patterns of the population densities appear in reaction-diffusion systems. However, this suggestion has not been proved completely, and thus the theories on reaction-diffusion systems have not been well developed yet.
Our objectives is to clarify how the following two dynamic patterns are generated in reaction-diffusion systems: the pattern with a corner layer; the stairs-like pattern every step of which moves forward with a different speed from each other. In the present investigation we prepare some basic theories which will be useful for the construction and analysis of asymptotic solutions which approximate to those two patterns.
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| Free Research Field |
非線形微分方程式、反応拡散系
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| Academic Significance and Societal Importance of the Research Achievements |
反応拡散系の解として出現することが数値的に予想されている動的パターンの存在を個別に証明する際に理論的な基盤となる「反応拡散近似」について、総合報告論文として出版することにより、反応拡散系に係る今後の漸近解析の発展に寄与できる。
副次的な結果として一例を示すと、常識的には近接相互作用である拡散とは何ら関係がないと思える遠隔相互作用さえも、拡散場での近接相互作用を記述する反応拡散系の特異極限として得られることが理論的に明らかになった。この事実は、反応拡散系を用いて解明できる未知の分野の裾野が意外と広大なことを暗示しているのでなかろうか。
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