Asympotic behaviors of spatial critical points and zeros of solutions of parabolic equations
Project/Area Number  10640175 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  Ehime University 
Principal Investigator 
SAKAGUCHI Shigeru (Hokkaido University, Faculty of School, Associate Professor), 理学部, 助教授 (50215620)

CoInvestigator(Kenkyūbuntansha) 
柳 重則 愛媛大学, 理学部, 助教授 (10253296)
MORIMOTO Hiroaki (Ehime University, Faculty of School, Professor), 理学部, 教授 (80166438)
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)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,800,000 (Direct Cost : ¥3,800,000)
Fiscal Year 1999 : ¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1998 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  heat equation / porous medium equation / initial boundary value problem / diffusion equation / spatial critical point / isothermal surface / interface / symmetry of domains / 熱方程式 / ポーラスメディアム方程式 / 初期境界値問題 / 拡散方程式 / 空間臨界点 / 等温面 / インターフェース / 領域の対称性 / 自己相似解 / 漸近挙動 / ピーラプラシアン / ノイマン境界条件 / リプシッツ領域 
Research Abstract 
(1) Level surfaces invariant with time of solutions of diffusion equations We consider solutions of the initialNeumann 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 LeviCivita (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 onedimensional porous medium equation We consider the Cauchy and the initialDirichlet problems for the onedimensional evolution pLaplacian 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 =
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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 initialDirichlet problem for the onedimensional evolution pLaplacian equation, we show that there exists a positive constant β=β(ρ) such that x(t)tィイD1β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 initialDirichlet 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 Output
(4results)