1999 Fiscal Year Final Research Report Summary
Analysis of Nonlinear Waves and Nonlinear Diffusions
| Project/Area Number |
09440066
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | TOKYO METOROPOLITAN UNIVERSITY |
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Principal Investigator |
MOCHIZUKI Kiyoshi Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (80026773)
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| Co-Investigator(Kenkyū-buntansha) |
KURATA Kazuhiro Tokyo Metropolitan University, Faculty of Sci., Associate Prof., 大学院・理学研究科, 助教授 (10186489)
SAKAI Makoto Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70016129)
ISHII Hitoshi Tokyo Metropolitan University, Faculty of Sci., Prof., 大学院・理学研究科, 教授 (70102887)
SUZUKI Ryuichi Kokushikan U., Faculty of Eng., Asso. Prof., 工学部, 助教授 (00226573)
KAWANAGO Tadashi Shizuoka U., Faculty of Eng., Associate Prof., 工学部, 助教授
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| Project Period (FY) |
1997 – 1999
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| Keywords | localized dissipation near infinity / Degenerate parabolic equation / Kirchhoff equation / Sturm-Liouvill operator / Schrodinger operator / Hamilton-Jacobi equation / Hele-Shaw fluid / bifurcation phenomena |
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| 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 1997-99, 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 blow-up and global existence of s
… More
olutions are shown to exist. Moreover, the critical blow-up, life span of blow-up 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 Sturm-Liouville 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 long-range potential.
Each investigator developed many important nonlinear problems, among which are included e.g., homogenization of the Hamilton-Jacobi equation, free boundary problem of Hele-Shaw flow, variational problems, stability of Navier-Stokes equations and nonlinear scattering. Less
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