| Project/Area Number |
16540107
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
SAKAMOTO Kunimochi Hiroshima University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (40243547)
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| Co-Investigator(Kenkyū-buntansha) |
MIMURA Masayasu Meiji University, Department of Science and Engineering, Professor, 理工学部, 教授 (50068128)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,200,000 (Direct Cost: ¥3,200,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | transition layer / interface / singular limit / asymptotic analysis / mean curvature flow / Lyapunov-Schmidt method / symmetry breaking bifurcation / reaction-diffusion system / 漸近展開 / 保存則 / 対流 / Lyapunov-Schmidt法 / 極小曲面 / 楕円型偏微分方程式 / 変分法 / 球対称 / 反応拡散対流系 / 界面方程式 / 分岐 / 安定性解析 |
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| Research Abstract |
In this research project, systems of reaction-diffusion (convection) equations are investigated from a viewpoint of singular limit analysis. The results obtained are summarized as follows.
1. In spherically symmetric multidimensional domains, reaction-diffusion systems of activator-inhibitor type are studied when the reaction rate of inhibitor is weak and the diffusion rate of activator is small. It is shown that symmetry breaking bifurcations of transition layer solutions occur as the diffusion rate of inhibitor is decreased. The bifurcation takes place infinitely often.
2. In activator-inhibitor systems of reaction-diffusion equations, when the reaction rates of both components are comparable and the diffusion rate of inhibitor is large, it is shown that symmetry breaking bifurcations of transition layer solutions occur as the diffusion rate of activator is decreased. The bifurcation takes place infinitely often. Moreover, the typical wave length in the direction parallel to the interf
… More
ace scales as proportional to the square root of the diffusion rate of activator.
3. Allen-Cahn equation is considered in three dimensional bounded domains, and stationary transition layer solutions whose interface intersects the domain boundary are studied. It is found that such solutions are possible only when the interface is a minimal surface intersecting the domain boundary in right angle. Moreover, the stability of such stationary transition layers is determined by an elliptic boundary value problem on the minimal surface with Robin type boundary conditions. For specific types of domains, construction of stationary transition layer solutions are carried out and their stability conditions are explicitly expressed in the form of computable quantities.
4. For scalar reaction-diffusion-convection equations of bi-stable type, asymptotic singular perturbation analysis is carried out to derive an interface equation which clearly displays the effects of convection on the motion of transition layers. It is suspected that the convection may stabilize stationary transition layers which without convection is know to be unstable. Less
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