1999 Fiscal Year Final Research Report Summary
Asympotic behaviors of spatial critical points and zeros of solutions of parabolic equations
| Project/Area Number |
10640175
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Ehime University |
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Principal Investigator |
SAKAGUCHI Shigeru (Hokkaido University, Faculty of School, Associate Professor), 理学部, 助教授 (50215620)
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| Co-Investigator(Kenkyū-buntansha) |
NAITO Manabu (Ehime University, Faculty of School, Professor), 理学部, 教授 (00106791)
HASHIMOTO Takahiro (Ehime University, Faculty of School, Research Associate), 理学部, 助手 (60291499)
JIMBO Shuichi (Hokkaido University, Graduate School of Science, Professor), 理学研究科, 教授 (80201565)
KISO Kazuhiro (Ehime University, Faculty of School, Professor), 理学部, 教授 (60116928)
MORIMOTO Hiroaki (Ehime University, Faculty of School, Professor), 理学部, 教授 (80166438)
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| Project Period (FY) |
1998 – 1999
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| Keywords | heat equation / porous medium equation / initial boundary value problem / diffusion equation / spatial critical point / isothermal surface / interface / symmetry of domains |
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| Research Abstract |
(1) Level surfaces invariant with time of solutions of diffusion equations
We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in Euclidean space, and with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita (1937) and Segre (1938), we classify the solutions whose isothermal surfaces are invariant with time. Furthermore, we can deal with nonlinear diffusion equations such as the porous medium equation, and we get similar classification theorems.
(2) Asymptotic behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equation
We consider the Cauchy and the initial-Dirichlet problems for the one-dimensional evolution p-Laplacian equation with p>1 for nonzero, bounded, and nonnegative initial data having compact support. It was shown that after a finite time the set of spatial critical points of the solution u in {u > 0} consists of one point, say x =
… More
x(t) for time t. In this research, we show that after a finite time x(t) is CィイD11ィエD1 in t. Furthermore, we can deal with generalized porous medium equations with sign changes, and we get CィイD11ィエD1 regularity of the interfaces with sign changes. Also, in the initial-Dirichlet problem for the one-dimensional evolution p-Laplacian equation, we show that there exists a positive constant β=β(ρ) such that x(t)tィイD1-βィエD1 tends to some positive constant as t → ∞.
(3) Stationary critical points of the heat flow and the symmetries of the domains
We consider the initial-Dirichlet problem for the heat equation on bounded and simply connected domains in the plane. By a new method with the help of the Riemann Mapping theorem in complex analysis, we give a characterization of domains invariant under the rotation of angle 2π/3 by making use of the stationary critical points of the heat flow. (Previously, only the characterizations of balls and centrosymmetric domains were obtained.) Furthermore, we consider stationary critical points of the heat flow in sphere SィイD1NィエD1 and in hyperbolic space HィイD1NィエD1, and prove several results corresponding to those in Euclidean space which have been proved in Magnanini and Sakaguchi (1997, 1999). Precisely. We get the characterizations of geodesic balls and centrosymmetric domains by making use of the stationary critical points of the heat flow. Less
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Research Products
(2 results)
