2004 Fiscal Year Final Research Report Summary
Structure of Solutions to partial differential equations degenerating on several hypersurfaces
| Project/Area Number |
15540188
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Osaka-Electro-Conmunication University |
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Principal Investigator |
MANDAI Takeshi Osaka-Electro-Communication University, Faculty of Engineering, Professor, 工学部, 教授 (10181843)
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| Co-Investigator(Kenkyū-buntansha) |
IGARI Katsuju Ehime University, Faculty of Engineeirng, Professor, 工学部, 教授 (90025487)
TAHARA Hidetoshi Sophia University, Faculty of Science and Engineering, Professor, 理工学部, 教授 (60101028)
ASAKURA Fumioki Osaka-Electro-Conmunication University, Faculty of Engineering, 工学部, 教授 (20140238)
SAKATA Sadahisa Osaka-Electro-Conmunication University, Faculty of Biomedical Engineering, 医療福祉工学部, 教授 (60175362)
YAMAHARA Hideo Osaka-Electro-Conmunication University, Faculty of Engineering, 工学部, 助教授 (30103344)
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| Project Period (FY) |
2003 – 2004
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| Keywords | degenerate p.d.e. / Fuchsian p.d.e. / Fuchsian variable / characteristic exponent / singular point / uniqueness / Cauchy problem / complex domain |
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| Research Abstract |
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.
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Research Products
(8 results)
