2001 Fiscal Year Final Research Report Summary
THE STUDY OF NONLINEAR PHENOMENA BY THE ASYMPTOTIC ANALYSIS
Project/Area Number 
11640124

Research Category 
GrantinAid for Scientific Research (C)

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

Research Institution  The University of Tokushima 
Principal Investigator 
ITO Masayuki The Univ. of Tokushima, Dept. of Math. & Natural Sc., Professor, 総合科学部, 教授 (70136034)

CoInvestigator(Kenkyūbuntansha) 
KODA Atsuhito The Univ. of Tokushima, Fac. of Technology, Associate Professor, 工学部, 助教授 (50116810)
FUKAGAI Yoshinobu The Univ. of Tokushima, Fac. of Technology, Associate Professor, 工学部, 助教授 (90175563)
NARUKAWA Kimiaki Naruto Edu. Univ., Fac. of School Ed., Professor, 学校教育学部, 教授 (60116639)
OHNUMA Masaki The Univ. of Tokushima, Dept. of Math. & Natural Sc., Lecturer, 総合科学部, 講師 (90304500)
MURAKAMI Koichi The Univ. of Tokushima, Dept. of Math. & Natural Sc., Associate Professor, 総合科学部, 助教授 (90219890)

Project Period (FY) 
1999 – 2001

Keywords  quasilinear / degenerate / elliptic equation / eigenvalue problem / bifurcation / comparison principle / difference equation / invariant curve 
Research Abstract 
The purpose of this project is to describe nonlinear phenomena mathematically by using asymptotic analysis. And we have the following results. 1) Narukawa and Fukagai proposed a mathematical model related to the nonlinear elasticity. This is described by a degenerate quasilinear elliptic equation whose principal part has different orders at 0 and at infinity. They have showed a global bifurcation diagram of positive solutions for a nonlinear eigenvalue problem of such quasilinear equations and, in particular, the coexistence of multiple positive solutions. These results obtained by regularity estimate of weak solutions and modifying the argument given by Ambrosetti, Brezis and Cerami in the semilinear case. 2) Ohnuma has investigated a class of singular degenerate parabolic equations including the pLaplace diffusion equation and the equation of the mean curvature flow, and proved the comparison principle for these equations. He also discovered a strong maximum principle of quasilinear degenerate elliptic equations. 3) Murakami showed a necessary and sufficient condition of the asymptotic stability of a fixed point for a higher order linear difference equation. He has also investigated some nonlinear difference equations, derived the formula to compute the stability conditions of the invariant curve caused by the NeimarkSacker bifurcation and, moreover, given the explicit expression of the invariant curve. 4) Kohda has obtained blowup criteria for a solution of an initial value problem of a semilinear parabolic equation. This is described by using a supersolusion and a subsolution of the stationary problem. Moreover, he had given some condition which guarantees the blowup of the solution.

Research Products
(12 results)