Study of the Mathematical structure of singularities

奇点的数学结构研究

• 批准号: 11214202
• 负责人: MATANO Hiroshi
• 金额: $ 7.04万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research on Priority Areas (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11214202/
• 关键词: singularity; singular perturbation method; singular limit; blow-up of solutions; nonlinear problems; dynamical systems; bifurcation theory; interface; 微分方程式; 非線型問題; 非線形偏微分方程式

(1) Singular limit of diffusion equations In certain kinds of diffusion equations, transition layers appear as the diffusion coefficients tends to zero. Matano has studied the singular limit for diffusion equations with spatially inhomogeneous coefficients. Yanagida has used this singular limit technique to proved the existence of stable periodic solutions. Funaki has investigated the behavior of interfaces that appear in equations with random deviation.(2) Blow-up in nonlinear heat equations Some blow-up solutions of nonlinear heat equations can be continued beyond the blow-up time in a certain sense. Matano has studied the dynamics of such blow-up solutions. Yanagida has obtained new formula for estimating the blow-up time.(3) Singularities in free boundary problems Weiss has applies his new method obtain the Hausdorff dimension of the singularity set of a 2-phase obstacle problem.(4) Singularities and bifurcation in comlex dynamical system Shishikura has obtained a new systematic method to study bifurcation problems and rigidity of Julia sets.

(1)扩散方程的奇异极限在某些扩散方程中，当扩散系数趋于零时，会出现过渡层。Matano研究了具有空间非齐次系数的扩散方程的奇异极限。柳田用这种奇异极限技巧证明了稳定周期解的存在性。Funaki研究了具有随机偏差的方程中出现的界面的行为。(2)在非线性热方程的爆破中，非线性热方程的一些爆破解在一定意义上可以在爆破时间之后继续存在。马塔诺研究了这种爆破解的动力学。柳田得到了估计爆破时间的新公式。(3)自由边界问题中的奇点Weiss应用他的新方法得到了两相障碍问题奇点集的Hausdorff维。(4)复杂动力系统中的奇点与分叉。Shishikura获得了一种新的系统方法来研究Julia集的分叉问题和刚性。

项目成果

M.Shishikura: "Bifurcation of parabolic fixed points"London Math. Soc. Lect. Note Ser.. 274. 325-363 (2000)

M.Shishikura：“抛物线不动点的分叉”伦敦数学。

Yanagida, Eiji: "Life span of solutions for a semilinear parabolic problem with small diffusion"J. Math. Ana. Appl.. 261. 350-368 (2001)

Yanagida, Eiji：“小扩散半线性抛物线问题解的寿命”J.

H.Matano: "Existence of L^1 connections between equilibria of a semilinear parabolic equation"J. Dynamics and Differential Equations. 14. 463-491 (2002)

H.Matano：“半线性抛物线方程的平衡点之间存在 L^1 连接”J。

T.Funaki: "Fluctuations for ▽φ interface model on a wall"Stoch.Proc.Appl.. 94. 1-27 (2001)

T.Funaki：“墙壁上 ▽φ 界面模型的波动”Stoch.Proc.Appl.. 94. 1-27 (2001)

G.Weiss,: "An obstacle-problem-like equation with two phases : pointwise regularity of the solution and an estimate of the Hausdorff dimension"Interfaces and Free Boundaries. 3. 1-8 (2001)

G.Weiss，：“具有两个阶段的类似障碍问题的方程：解的逐点正则性和豪斯多夫维数的估计”接口和自由边界。