Analysis of the structure of solutions of nonlinear partial differential equations
Project/Area Number  07454031 
Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  University of Tokyo 
Principal Investigator 
MATANO Hiroshi University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院・数理科学研究科, 教授 (40126165)

CoInvestigator(Kenkyūbuntansha) 
IWASAKI Katunori University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (00176538)
MATUMOTO Yukio University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20011637)
KUSUOKA Shigeo University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (08640254)
OCHIAI Takushiro University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90028241)
TSUTSUMI Yoshio University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (10180027)

Project Period (FY) 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1996 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  nonlinear partial differential equations / qualitatuve theory / diffusion equations / infinite dimensional dynamical systems / parabolic equations / attractor / パターン形成 / 定性的研究 / 境界現象 
Research Abstract 
Westudied the structure of the infinite dimensional dynamical systems defined by nonlinear parabolic equations in one space dimension and showed that their global attractors are always finite dimensional manifolds. This result implies that the essential features of the longtime behavior of solutions can be described by a finite system of ordinary differential equations, and is therefore important from the point of view of qualitative theory. It should be noted that the socalled inertial manifold theory, which has been wellknown since mid 1980's as a tool for studying the finite dimensionality of attractors, does not apply to the equation treated in our research. 2. We studied the behavior of solutions of degenerate diffusion equations and proved that any unstable equilibrium solution has an unstable manifold of infinite Hausdorff dimension. This result shows that there is essential difference between the dynamical structure of degenerate diffusion equations and that of nondegenerate equations. currently Matano is also studying the properties of traveling waves for nonlinear diffusion equations with spatially priodic coefficients and has obtained partial results. 3. We studied a mathematical model which combines Maxwell equation and Schrodinger equation. We obtained new results on the uniqueness and global existence of solutions. 4.Kusuoka, one of the investigaters of the research project, has been studying problems in mathematical finance with probabilistic method. He has obtained some interesting results.

Report
(3results)
Research Output
(17results)