2009 Fiscal Year Final Research Report
Qualitative Properties of Solutions of Differential Equations Modeling Biological Pattern Formation
| Project/Area Number |
18204010
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Tohoku University |
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Principal Investigator |
TAKAGI Izumi Tohoku University, 大学院・理学研究科, 教授 (40154744)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGIDA Eiji 東北大学, 大学院・理学研究科, 教授 (80174548)
IKEDA Hideo 富山大学, 大学院・理工学研究部, 教授 (60115128)
NAGASAWA Takeyuki 埼玉大学, 大学院・理工学研究科, 教授 (70202223)
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| Co-Investigator(Renkei-kenkyūsha) |
IIDA Masato 宮崎大学, 工学部, 教授 (00242264)
ISHIGE Kazuhiro 東北大学, 大学院・理学研究科, 准教授 (90272020)
UEYAMA Daishin 明治大学, 理工学部, 准教授 (20304389)
OGAWA Takayoshi 東北大学, 大学院・理学研究科, 教授 (20224107)
MOCHIZUKI Atsuhi 理化学研究所, 基礎研究所, 主任研究員 (10304726)
YAMADA Sumio 東北大学, 大学院・理学研究科, 准教授 (90396416)
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| Project Period (FY) |
2006 – 2009
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| Keywords | 反応拡散方程式 / 活性因子-抑制因子系 / 進行波解 / 非一様な媒質 / 幾何学的変分問題 / 曲線の運動 |
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| Research Abstract |
Collapse of patterns is a newly found phenomenon characteristic to some reaction-diffusion systems possessing singular nonlinearities, where patterns are formed at first but eventually converge to a nonregular steady state. We have given sufficient conditions for patterns to collapse and also for solutions to blow-up in finite time. In addition, qualitative properties of solutions such as the dynamics of maximum points and/or asymptotic forms of solutions have been studied in detail.
Moreover, movement of planar closed curves driven by bending energy is considered as a lower dimensional analogue for the geometric variational problem which determines the shape of red blood cells. All the critical points of the energy functional under some constraints are found and the gradient flow of the constraint minimization problem has been constructed.
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