Project/Area Number 
12640225

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  Ryukoku University 
Principal Investigator 
YOTSUTANI Shoji Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)

CoInvestigator(Kenkyūbuntansha) 
MORITA Yoshihisa Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (10192783)
NINOMIYA Hirokazu Ryukoku Univ., Faculty of Sci. and Tech., Associate Professor, 理工学部, 助教授 (90251610)
YANAGIDA Eiji Tohoku Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
IKEDA Tsutomu Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (50151296)
OKA Hiroe Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)

Project Period (FY) 
2000 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  elliptic equation / radial solution / canonical form / crossdiffusion / reaction diffusion / nonlocal / 半線形楕円形方程式 / スカラー曲率方程式 / 不完全分岐 / 反応拡張方程式 / 進行波解 
Research Abstract 
First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida. Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations. Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida. As related problems, we are investigating a limiting equation to a crossdiffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.
