1990 Fiscal Year Final Research Report Summary
Study of Nonlinear Partial Differential Equations from the View Point of Functional Analysis
| Project/Area Number |
01460004
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | University of Tokyo |
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Principal Investigator |
MASUDA Kyuya University of Tokyo, Faculty of Science, Pfofessor, 理学部, 教授 (10090523)
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| Co-Investigator(Kenkyū-buntansha) |
SHOJI Mayumi University of Tokyo, Faculty of Science, Assistant, 理学部, 助手 (10216161)
ISHIMURA Naoyuki University of Tokyo, Faculty of Science, Assistant, 理学部, 助手 (80212934)
FUKAYA Kenji University of Tokyo, Faculty of Science, Associate Professor, 理学部, 助教授 (30165261)
KATAOKA Kiyomi University of Tokyo, Faculty of Science, Associate Professor, 理学部, 助教授 (60107688)
MATANO Hiroshi University of ToKYO, Faculty of Science, Associate Professor, 理学部, 助教授 (40126165)
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| Project Period (FY) |
1989 – 1990
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| Keywords | reaction-diffusion system / nonlinear heat equation / blow-up of solutions / bifurcation theory / Riemannian manifold / water wave |
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| Research Abstract |
We have studied nonlinear partial differential equations from many aspects in the period 1989 May-1991 March. Masuda succeeded first in showing the existence of stable periodic solutions of reaction-diffusion systems of some type, the most important class of nonlinear parabolic equations. He also considered the heat equation in the form u=*u+a (x) u^p and showed any non-negative solution blows up in a finite time if a (x) is posive at some point in the domain considered. Matano considered the nonlinear heat equation in some one-dimensional space interval with Dirichlet boundary condition.
He could show the remarkable result that any solutions with a continuous function as the initial data blows up at a (at most) finite number of points.Ishimura showed the existence and nonexistence of multi-valued solutions of nonlinear elliptic equations of the form *u=u^p. shouji made research into the permanent progressive waves and analyzed the bifurcation of progressive waves. Fukaya made research into nonlinear differential equations in differential geometry. He clarified the structure of small ball in the Riemannian manifold with sectional curvature less than one in absolute value.
Kataoka succeeded in showing the hypoellipticity of degenerate elliptic equations.
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