Project/Area Number  09440046 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
解析学

Research Institution  TOHOKU UNIVERSITY 
Principal Investigator 
TAKAGI Izumi Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (40154744)

CoInvestigator(Kenkyūbuntansha) 
中野 史彦 東北大学, 大学院・理学研究科, 助手 (10291246)
IIDA Masato Faculty of Education, Iwate University, Lecturer, 教育学部, 講師 (00242264)
NAGASAWA Takeyuki Graduate School of Science, Tohoku University, Associate Professor, 大学院・理学研究科, 助教授 (70202223)
NISHIURA Yasumasa Research Institute for Electronic Science, Hokkaido University, Professor, 電子科学研究所, 教授 (00131277)
増田 久弥 東北大学, 大学院・理学研究科, 教授 (10090523)
TSUTSUMI Yoshio Graduate School of Science, Tohoku University, Professor, 大学院・理学研究科, 教授 (10180027)
TACHIZAWA Kazuya Graduate School of Science, Tohoku University, Lecturer, 大学院・理学研究科, 講師 (80227090)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥10,900,000 (Direct Cost : ¥10,900,000)
Fiscal Year 1999 : ¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1998 : ¥3,600,000 (Direct Cost : ¥3,600,000)
Fiscal Year 1997 : ¥4,000,000 (Direct Cost : ¥4,000,000)

Keywords  reactiondiffusion system / singular perturbation / spikelayer solutions / transition layer / stability / 反応拡散方程式 / 特異摂動 / スパイク解 / 遷移層 / 安定性 / 反応拡散系 / 集中現象 / 半線型楕円型偏微分方程式 / パターン形式 / 遷移過程 / 活性因子 
Research Abstract 
1. Takagi considered the construction and stability of stationary solutions of a reactiondiffusion system of activatorinhibitor type. With the cooperation of WeiMing Ni and Eiji Yanagida, he proved the following in the case of one dimensional domains : (i) The existence of stationary solutions concentrating at the boundary point when the activator diffuses slowly and the inhibitor diffuses very fast. (ii) If the relaxation parameter of the inhibitor reaction is small then these solutions are stable ; while they are unstable if the relaxation parameter is sufficiently large. (iii) A oneparameter family of periodic solutions concentrating around the boundary point bifurcates from the stationary solution. Moreover, these results are generalized to higher dimensional domains in the case where the diffusion rate of the inhibitor is infinite. 2. Nishiura and Iida studied the behavior of solutions to the initialboundary value problem for reactiondiffusion systems which generate sharp transition layers. Nishiura established a theory to explain the mechanism of selfreplicating patterns. Iida constructed a reactiondiffusion system whose singular limit reduces to the classical Stefan problem. 3. Tsutsumi, Tachizawa and Nakano studied Schroedinger equations by applying techniques in real analysis. They obtained new results on the wellposedness of the initial value problem, and on the asymptotic distribution of eigenvalues. 4. Masuda and Nagasawa considered mainly the behavior of solutions to nonlinear diffusion equations. Masuda proved the maximum principle for weak solutions. Nagasawa refined the energy inequality for weak solutions to the NavierStokes equations.
