1989 Fiscal Year Final Research Report Summary
Properties of Solutions of Partial Differential Equations and Their Applications
| Project/Area Number |
63540134
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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 Professor, 総合科学部, 教授 (40079033)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Masako Professor, 総合科学部, 教授 (50081419)
ISHII Noburo Associate Professor, 総合科学部, 助教授 (30079024)
KONNO Yasuko Associate Professor, 総合科学部, 助教授 (70028231)
TANIGUCHI Kazuo Lecturer, 総合科学部, 講師 (80079037)
SHINKAI Kenzo Professor, 総合科学部, 教授 (50079034)
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| Project Period (FY) |
1988 – 1989
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| Keywords | hyperbolic equation / Fourier integral operator / propagation of singularity / Cauchy problem / cohomology / unitary representation / class field / lattice path combinatorics |
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| Research Abstract |
The fundamental solution of the Cauchy problem for a hyperbolic operator is given in the form of Fourier integral operator. As shown below, when the problem is not C" well-posed, the symbol of the fundamental solution has exponential growth, that is, it is estimated not only from above but also from below by (1) C exp[cxi^<1/k>], c > 0, The constant kappa in (1) corresponds to the constant in the necessary and sufficient condition for the well-posedness in Gevrey classes. In order to study this phenomena we define UWF^<{mu}>(u)(ultra wave front sets) for u that belongs to the space of ultradistributions S{k}' by (chi_0,xi_0) <not a member of> UWF^<{mu}>(u) <tautomer> *_<epsilon> > O*C ; |X^u(xi)| <less than or equal> exp[epsilon < xi >^<1/mu>], where X * S{k}*C^*_ and xi belongs to a conic neighborhood of xi_0. Then by using UWP^<{mu}>(u) we can state the propagation of very high singularities for the solution of not C^* well-posed Cauchy problem. We also construct the fundamental solutions of the Cauchy problem for degenerate hyperbolic operators (2) L_1 = THETA^2_ - t^2_ - at^kTHETA^x with 0 < k < j - 1 and (3) L_2 = THETA^2_ - x^<2j>THETA^2_ - aTHETA^x with an even integer j and we investigated other related topics.
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