2023 Fiscal Year Final Research Report
Elucidation of structure of solutions to chemotaxis systems with non-linear sensitivity functions
| Project/Area Number |
18K03386
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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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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Fukuoka University |
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Principal Investigator |
Senba Takasi 福岡大学, 理学部, 教授 (30196985)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 走化性方程式系 / 知覚関数 / 時間大域的存在 / 爆発 |
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| Outline of Final Research Achievements |
We get some results on time-global existence and finite-time blowup of solutions to chemotaxis systems with nonlinear sensitivity functions. In particular, we concentrated our study in the case where sensitivity functions are positive constants times the logarithmic function. We clarify that those chemotaxis system do not have blowup solutions, if constants of logarithmic sensitivity functions are less than a certain value. We say the value threshold number. In this study, we use solutions to a simplified system as auxiliary functions. This argument is found by our research.
On blowup of solutions, it is known that those systems have finite-time blowup solutions if constants of logarithmic sensitivity functions are bigger than 2 times threshold number. We construct finite-time blowup solutions to those systems also in the case where the constant is a certain value between threshold number and 2 times threshold number.
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| Free Research Field |
偏微分方程式論
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| Academic Significance and Societal Importance of the Research Achievements |
線形の知覚関数を持つ走化性方程式系の解の性質に関しては解の時間大域的存在及び爆発の両面から研究が進んでおり、リアプノフ関数がその研究に重要な役割を果たしている。一方、非線形知覚関数を持つ走化性方程式系のリアプノフ関数は発見されておらず、そのため研究は遅れており、その研究には今まで用いられた手法とは別の手法の発見が重要であった。本研究に用いられた補助関数を用いる手法によって非線形知覚関数を持つ走化性方程式系の研究が進展することが期待できる。さらに、走化性方程式系は生物学的な現象を背景としており、非線形知覚関数の中でも対数関数は生物学のモデルの中で重要な位置づけとなっている。
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