Asympotic behaviors of spatial critical points and zeros of solutions of parabolic equations

抛物方程空间临界点和解零点的渐近行为

• 批准号: 10640175
• 负责人: SAKAGUCHI Shigeru
• 金额: $ 2.43万
• 依托单位: Ehime University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640175/
• 关键词: heat equation; porous medium equation; initial boundary value problem; diffusion equation; spatial critical point; isothermal surface; interface; symmetry of domains; 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

(1) Level surfaces invariant with time of solutions of diffusion equationsWe 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 equationWe 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 domainsWe 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

（1）与扩散方程的解决方案的水平表面不变，我们考虑欧几里得空间中有界Lipschitz域上热量方程的最初 - Neumann问题的解决方案，并借助Isoparametric Hypersurfaces的分类定理，分类为Levi-Civita（19337）和Segivita（1933）（1933）（1933）（1933）等温表面随时间不变。 Furthermore, we can deal with nonlinear differences equations such as the porous medium equation, and we get similar classification theorems.(2) Asymmetric behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equationWe 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具有紧凑支持的数据。结果表明，在有限的时间之后，{u> 0}中解决方案u的空间临界点集由一个点组成，例如时间t x =…更多x（t）。在这项研究中，我们表明有限的时间x（t）为t中的CII D11。此外，我们可以处理具有符号变化的通用多孔培养基方程，并且可以通过符号更改获得界面的CII D11 D1规律性。同样，在一维进化P-拉平方程的初始 - 迪里奇问题中，我们表明存在正常常数β=β（ρ），使得x（t）tii d1-βied1趋向于t→∞趋向于t→∞。平面中的域。通过在复杂分析中借助Riemann映射定理的新方法，我们通过利用热流的固定临界点在角度2π/3的旋转下对域不变。 （以前，仅获得了球和中心对称域的特征。）此外，我们考虑了球体中的热流的固定临界点SIE D1NIE D1和多功能空间H1nie D1，并证明了几个结果，与euclidean空间中的那些结果相对应，这些结果已在Magnanini和Sakaguchi（19999999年）中得到证明。恰恰。我们通过利用热流的固定临界点来获得大地球和中心对称域的特征。较少的

项目成果

Shigeru Sakaguchi: "When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?"Journal d'Analyse Mathematique. Vol.78. 219-243 (1999)

Shigeru Sakaguchi：“扩散方程解的空间水平面何时相对于时间变量不变？”Journal dAnalyse Mathematique。

Shigeru Sakaguchi: "When are the spatial level surfaces of solutions of diffusion equations in variant with respect to the time variable?"Journal d' Analyse Mathematique. 78. 219-243 (1999)

Shigeru Sakaguchi：“扩散方程解的空间水平面何时相对于时间变量发生变化？”《分析数学杂志》。

Shigeru Sakaguchi: "When are the spatial level surfaces of solutions of diffusion equations invarlant with respect to the time variable?"Journal d' Analyse mathematique. 78. 219-243 (1999)

Shigeru Sakaguchi：“什么时候扩散方程解的空间水平面相对于时间变量是不变的？”《分析数学杂志》。

Shigeru Sakaguchi: "When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?" J.Analyse Math.,. 印刷中.

Shigeru Sakaguchi：“扩散方程解的空间水平面何时相对于时间变量不变？” J.Analyse Math.，正在出版。