Structure of the Solutions to Partial Differential Equations Degenerating on the Initial Surface, and its Applications

初表面上退化的偏微分方程解的结构及其应用

• 批准号: 12640194
• 负责人: MANDAI Takeshi
• 金额: $ 2.24万
• 依托单位: Osaka Electro-Communication University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640194/
• 关键词: Fuchsian partial differential equations; regular singularity; method of Frobenius; characteristic exponent; characteristic Cauchy problem; フックス型偏微分作用素; フックス型変数; Volevicタイプシステム

The indicial polynomial and its zeros called characteristic exponents play an important role in the study of Fuchsian partial differential equations in the sense of Baouendi-Goulaouic, that is, linear partial differential equations with regular singularity along the initial surface. We had succeeded to construct a solution map which gives the local structure of the solutions to homogeneous single Fuchsian partial differential equations and first order Fuchsian systems in a complex domain, without any assumption on the indicial polynomial.In this research, we constructed a solution map for Fuchsian systems of Volevic type (not necessarily first order). We need to consider the systems without reducing them to first order systems, since such a reduction involves singular transformations.We also considered partial differential equations with several Fuchsian variables. We constructed distribution null-solutions for such equations in the real domain. The situation is far more complicated than that for equations with a single Fuchsian variable, for which we had already constructed distribution null-solutions.

在Baouendi-goulaouic的意义上，指示性多项式及其称为特征指数的零在研究紫色部分微分方程的研究中起着重要作用，即，线性部分微分方程在初始表面具有正常的奇异性。我们成功地构建了一个解决方案图，该图为均质的单个紫色偏微分方程和一阶在复杂域中的fuchsian系统提供了解决方案的局部结构，而无需对指示多项式进行任何假设。在这项研究中，我们构建了一个解决方案图。对于Volevic类型的Fuchsian系统（不一定是一阶）。我们需要考虑这些系统而不将它们减少为一阶系统，因为这样的减少涉及奇异转换。我们还认为具有几个Fuchsian变量的偏微分方程。我们在真实域中为此类方程构建了分布空分。这种情况要比单个Fuchsian变量的方程式要复杂得多，我们已经为此构建了分布空分。

项目成果

T. Hara and S. Sakata: "Dynamics of a delay differential system with periodically oscillatory coefficients"Nonlinear Analysis. 47. 4399-4408 (2001)

T. Hara 和 S. Sakata：“具有周期性振荡系数的延迟微分系统的动力学”非线性分析。

J.E.C. Lope and H. Tahara: "On the analytic continuation of solutions to nonlinear partial differential equations"J. Math. Pures Appl.. 81. 811-826 (2002)

J.E.C.

Lope, J.E.C., Tahara, H.: "On the analytic continuation of solutions to nonlinear partial differential equations"J. Math. Pures Appl.. 81. 811-826 (2002)

Lope, J.E.C., Tahara, H.：“关于非线性偏微分方程解的解析延拓”J.

H. Chen, Z. Luo and H. Tahara: "Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity"Ann. Inst. Fourier, Grenoble. 51. 1599-1620 (2001)

H. Chen、Z. Luo 和 H. Tahara：“具有不规则奇点的非线性一阶全特征型偏微分方程的形式解”Ann。

Mandai,T.: "The Method of Frobenius to Fuchsian Partial Differential Equations "J.Math.Soc.Japan. 52:3. 645-672 (2000)

Mandai,T.：“Frobenius 求解 Fuchsian 偏微分方程的方法”J.Math.Soc.Japan。