1998 Fiscal Year Final Research Report Summary
Study of structure of solution to partial differential equations
| Project/Area Number |
09640221
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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 |
解析学
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| Research Institution | Sophia University |
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Principal Investigator |
OUCHI Sunao Sophia Univ.Fac.Science and Technology, professor, 理工学部, 教授 (00087082)
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| Co-Investigator(Kenkyū-buntansha) |
GOTO Satoshi Sophia Univ.Fac.Science and Technology, assistant, 理工学部, 助手 (00286759)
HIRATA Hitoshi Sophia Univ.Fac.Science and Technology, assistant, 理工学部, 助手 (20266076)
YOSHINO Kunio Sophia Univ.Fac.Science and Technology, lecturer, 理工学部, 講師 (60138378)
TAHARA Hidetoshi Sophia Univ.Fac.Science and Technology, professor, 理工学部, 教授 (60101028)
UCHIYAMA Koichi Sophia Univ.Fac.Science and Technology, professor, 理工学部, 教授 (20053689)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Partial differential equations in the complex domain / Asymptotic expansion / singularity |
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| Research Abstract |
885352We study partial differential equation P(z, delta^<alpha>u) = 0 in the complex do-main, where u(z) admits singularities on the complex hypersurface {z_0 = O}. The results are the following :
1. Let P(z, delta)u(z) = f(z) be a linear partial differential equation, where f(z) has also admits singularities on K.We show under some con-ditions on P(z, delta) that there is an exponent gamma > 0 such that if u(z) grows at most some exponential order near z0 0, that is, for any epsilon > 0, |u(z)| <less than or equal> C_<epsilon> exp(epsilon|z_0|^<-gamma>) and f(z) has a Gevrey type asymptotic expansion f(z) = SIGMA^^+=__0fn(z^l) * 0 in a sectorial region, where |fn(z^l)| <less than or equal> AB^nGAMMA(n/gamma+1), then u(z) has also an asymptotic expansion like f(z) as z_0 tends to 0.
2. The existence of solutios of formal power series of z_0, whose coefficients have Gevrey type estimate.
3. The uniqueness of singular solutions in some class of functions for some nonlinear partial differential equations.
We also study Davey-Stewartson equation which is one of nonlinear Schrodinger type equations and obtain the global existence of the initial value problem with small initial data and decay estimates for this equation.
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