| Project/Area Number |
60540116
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | University of Osaka Prefecture |
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Principal Investigator |
OKANO Hatsuo Department of Mathematics, University of Osaka Prefecture, 総合科学部, 教授 (40079033)
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| Co-Investigator(Kenkyū-buntansha) |
TOZAKI Yoshiharu Department of Mathematics, University of Osaka Prefecture, 総合科学部, 助手 (70079036)
TANIGUCHI Kazuo Department of Mathematics, University of Osaka Prefecture, 総合科学部, 講師 (80079037)
SHINKAI Kenzo Department of Mathematics, University of Osaka Prefecture, 総合科学部, 教授 (50079034)
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| Project Period (FY) |
1985 – 1986
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| Project Status |
Completed (Fiscal Year 1986)
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| Budget Amount *help |
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1986: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1985: ¥900,000 (Direct Cost: ¥900,000)
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| Keywords | Pseudo-differential operator / Fourier integral operator / Weakly hyperbolic equation / Stokes coefficient / Gevrey symbol / Multi-product / 多重積 / 特異性の伝播 / 準楕円性 / 波面集合の分岐 |
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| Research Abstract |
We investigated properties of Fourier integral operators and studied the singularities of solutions of weakly hyperbolic operators. We clarified the Gevrey wave front sets of solutions of differential equations with coefficients in Gevrey classes. The results are as follows:
1. The appearance of wave front sets of fundamental solutions of weakly hyperbolic operators depends on their lower order terms and their fundamental solutions may have the Gevrey wave front sets. In our project we construct exactly the fundamental solutions and investigate their wave front sets and Gevrey wave front sets. Then we can show that the solutions of the Cauchy problem have the branching singularities in a Gevrey class. In order to construct the fundamental solutions we use the Stokes coefficients of associated ordinary differential equations. Then, we can determine exactly whether or not the solution of the Cauchy problem of a degenerate hyperbolic equation has branching singularities.
2. In the study of singularities of solutions of differential equations with coefficients in Gevrey classes we give a precise estimate of multi-products of Fourier integral operators and pseudo-differential operators. We construct, as an application, the fundamental solution for a weakly hyperbolic operator and the parametrix of a hypoelliptic operator, and investigate the Gevrey wave front sets of solutions. Throughout this project we use mainly oscillatory integrals.
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