2023 Fiscal Year Final Research Report
Relation between spatial heterogeneity and nonlocality for pattern dynamics of reaction-diffusion systems
| Project/Area Number |
19K14588
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Institute of Physical and Chemical Research (2022-2023)
The University of Tokyo (2019-2021) |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 反応拡散系 / 非局所反応拡散方程式 |
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| Outline of Final Research Achievements |
In this study, we studied the relation between patterns and spatial heterogeneity or nonlocality for reaction-diffusion systems with spatial heterogeneity or nonlocality. For the spatial heterogeneity, we treated spike-shaped solutions called concentration phenomena and proved that the location of the maximum converges to the critical point of the locator function consisting of the coefficients of the equation.
We also approximated solutions of reaction-diffusion equations involving convolution integrals by the first component of solutions to reaction-diffusion systems. We proved the reaction-diffusion approximation of the nonlocal reaction-diffusion equation in the case that the kernel of the convolution integrals is a general continuous function. We considered stability problems for solutions of these reaction-diffusion systems and proved that the Evans functions can be constructed.
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| Free Research Field |
非線型偏微分方程式
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| Academic Significance and Societal Importance of the Research Achievements |
空間的非一様性を含む反応拡散方程式に対する点凝集現象に関する研究は，数学的にも応用上も重要である．生物の発生過程において，幾何学的な情報よりも環境の非一様性の方が影響が大きいことを表している．また，非局所反応拡散方程式に対して，領域全体での積分を含むので，従来の解析法を使うことができない場合があり，新たな解析法の確立が必要である．非局所反応拡散方程式の反応拡散近似は新たな解析法の一つであり，解の挙動や安定性を調べる時に有用と考えられる．Evans関数の構築は，様々な進行波解，例えば2つの進行波を組み合わせた進行波の安定性解析にも適用可能であり，汎用性が高炒め有用であると考えられる．
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