Structure of Solutions to partial differential equations degenerating on several hypersurfaces

多个超曲面上退化的偏微分方程解的结构

基本信息

批准号：
15540188
负责人：
MANDAI Takeshi
金额：
$ 1.86万
依托单位：
Osaka-Electro-Conmunication University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

The indicial polynomial and its zeros called characteristic exponents play important roles 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. In these equations, a variable (t) is treated as a special variable, which is called the Fuchsian variable. In 2004, we showed the existence of distribution null-solutions having their supports on the specific quadrant for equations with several Fuchsian variables (in the sense of N.S.Madi) under natural assumptions, with the collaboration with Professor M.Mechab and Ms.M.Belarbi in Algeria. In the process, it became clearer that the situation of the existence of null-solutions in the case of several Fuchsian variables is very different from that in the case of one Fuchsian variable. In 2005, we considered the problem concentrating on those points. We could not obtain good result as we had expected, we could clarify the relation between our result and the existence result of null-solutions to the degenerate weakly hyperbolic equations, which had been formerly obtained by us. Further progress is expected on the ground of this relation.We could obtain also the following results and others.1.Determination of the structure of singularities of solutions to nonlinear Fuchsian partial differential equations without any additional assumptions on the characteristic exponents. 2. A very sharp result on necessary conditions and sufficient conditions for first order nonlinear partial differential equations of normal form in a complex domain in order to have singular solutions. 3. For nonlinear partial differential equations called "nonlinear totally characteristic type" in complex domains, 1)the existence of singular points, 2)the nonexistence of singular points, 3)the uniqueness of solutions.

指示性多项式及其称为特征指数在Baouendi-goulaouic的意义上，即线性偏微分方程在初始表面具有正常奇异性的线性偏微分方程，在Baouendi-goulaouic的意义上起着重要作用。在这些方程式中，变量（t）被视为特殊变量，称为紫色变量。在2004年，我们展示了分布零分解的存在，在自然假设下，在具有多个Fuchsian变量（在N.S.Madi的意义上）的方程方程中对特定象限进行了支持，并与M.Mechab和M.Mechab教授和女士的合作进行了合作。阿尔及利亚的belarbi。在此过程中，很清楚，在几个紫红色变量的情况下，存在无溶液的情况与一个fuchsian变量的情况大不相同。在2005年，我们考虑了关注这些要点的问题。我们无法像预期的那样获得良好的结果，我们可以阐明结果与否定结果与堕落的弱肥大方程的存在之间的关系，这是我们以前获得的。从这种关系的角度来看，您还可以获得以下结果和其他结果。1。确定对非线性紫外线偏微分方程的解决方案的结构，而没有任何特征指数的其他假设。 2。在复杂域中正常形式的一阶非线偏微分方程的必要条件和足够条件的非常尖锐的结果，以便具有奇异的溶液。 3。对于复杂域中的非线性部分微分方程，称为“非线性完全特征类型”，1）单数点的存在，2）单数点不存在，3）解决方案的独特性。