Project/Area Number  08640223 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Nagasaki Institute of Applied Science 
Principal Investigator 
KAJIKIYA Ryuji Nagasaki Institute of Applied Science, Faculty of Engineering, Professor, 工学部, 教授 (10183261)

CoInvestigator(Kenkyūbuntansha) 
SENBA Takasi Miyazaki University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30196985)

Project Fiscal Year 
1996 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,300,000 (Direct Cost : ¥2,300,000)
Fiscal Year 1998 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1997 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1996 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  nonlinear elliptic equation / group invariant solution / variational method / blowup / parabolic system / chemotaxis / 非線形楕円型方程式 / 変分法 / parabelic system / 走化性 / 生物モデル 
Research Abstract 
1. We study the EmdenFowler equation, which is one of partial differential equations of elliptic type, in a ball or annulus of <planck's constant>dimensional Euclid space. Let C be a closed subgroup of the orthogonal group 0(<planck's constant>). A solution mu(x) of the equation is called G invariant if mu is invariant under G action. Any radial solution becomes G invariant. The converse problem is considered. The group G is a transformation group on the unit sphere because G is a subgroup of the orthogonal group. We prove that there exists a G invariant nonradial solution if and only if G is not transitive on the unit sphere. This result is proved by using variational method, functional analysis, Lie transformation group and SturmLiouville theory of ordinary differential 2. We study the KellerSegel equation which is a mathematical model to describe a cellular slime having the oriented movement. (1)A parabolic system which is a simplification of the KellerSegel equation is considered. When a sensitive function is linear and the space dimension is two, the asymptotic behavior of a blowup solution is investigated in detail. (2)It is proved that a radial solution blows up as its L^1 density concentrates at the origin. (3)The L^1 total mass of a solution is chosen as a parameter. Then the existence and non existence results of nonconstant stationary solutions are obtained by using the parameter.
