Project/Area Number  09440066 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
解析学

Research Institution  TOKYO METOROPOLITAN UNIVERSITY 
Principal Investigator 
MOCHIZUKI Kiyoshi Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (80026773)

CoInvestigator(Kenkyūbuntansha) 
SUZUKI Ryuichi Kokushikan U., Faculty of Eng., Asso. Prof., 工学部, 助教授 (00226573)
肥田野 久二男 東京都立大学, 大学院・理学研究科, 助手 (00285090)
SAKAI Makoto Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70016129)
KURATA Kazuhiro Tokyo Metropolitan University, Faculty of Sci., Associate Prof., 大学院・理学研究科, 助教授 (10186489)
ISHII Hitoshi Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70102887)
川中子 正 静岡大学, 工学部, 助教授
KAWANAGO Tadashi Shizuoka U., Faculty of Eng., Associate Prof.

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥8,000,000 (Direct Cost : ¥8,000,000)
Fiscal Year 1999 : ¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1998 : ¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1997 : ¥2,800,000 (Direct Cost : ¥2,800,000)

Keywords  localized dissipation near infinity / Degenerate parabolic equation / Kirchhoff equation / SturmLiouvill operator / Schrodinger operator / HamiltonJacobi equation / HeleShaw fluid / bifurcation phenomena / 局在する摩擦項 / 退化放物型方程式 / Kirchhoff方程式 / SturmLiouirlle作用素 / Schrodinger作用素 / HamiltonJacobi方程式 / HeleShaw流 / 分岐現象 / KPP方程式 / SturmLiouville作用素 / HaleShaw流 / FeffermanPhong不等式 / 非線形散乱 / 解の爆発 / 反応拡散方程式 / GinzburgLandau方程式 / エネルギー減衰 / Lifespan / 波動方程式 / KcrichhoH方程式 / 追化放物型方程式 / 爆発問題 / 漸近挙動 
Research Abstract 
Among various kind of problems on differential equations, in this project, we are mainly concerned with those related to the Applied Mathematics, Physics and Technology. Summarizing the results obtained by the investigators in the period 199799, we can say that the objective of this project is accomplished fruitfully. The head investigator published 9 papers analyzing the nonlinear waves and nonlinear diffusions. The topics include the following: (1) Decay and asymptotics of nonlinear waves: The existence of the scattering state is proved for acoustic wave equations with nonlinear dissipation. The existence of global small solution and its energy decay are established for the Kirchhoff equation (describing the vibration of elastic string) with dissipation localized near infinity. (2) Semilinear or quasilinear degenerate parabolic equations: Reaction diffusion systems with nonlinear power source term are considered. The critical exponents which divide the blowup and global existence of s
… More
olutions are shown to exist. Moreover, the critical blowup, life span of blowup solutions and the asymptotic behavior for time goes to infinity of global solutions are proved. Similar results are also obtained for the quasilinear equations describing fluids in porous media and combustion process in plasma. (3) Spectral and scattering theory: A new formulation and results are obtained for the spectral inverse problem for the classical SturmLiouville operator. Scattering theory is established for the wave equation with small dissipation or hoarding. Moreover, by generalizing the former results, we expanded the applicability of the principle of limiting absorption for the Schrodinger operator oscillating longrange potential. Each investigator developed many important nonlinear problems, among which are included e.g., homogenization of the HamiltonJacobi equation, free boundary problem of HeleShaw flow, variational problems, stability of NavierStokes equations and nonlinear scattering. Less
