1998 Fiscal Year Final Research Report Summary
THE MATHEMATICAL ANALYSIS TO NON-LINEAR PHENOMENA THROUGH NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS
| Project/Area Number |
09640276
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokushima |
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Principal Investigator |
ITO Masayuki The Univ.of Tokushima, Dept.of Math.& Natural Sc., Associate P., 総合科学部, 助教授 (70136034)
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| Co-Investigator(Kenkyū-buntansha) |
KODA Atsuhito The Univ.of Tokushima, Fac.of Technology, Associate p., 工学部, 助教授 (50116810)
MURAKAMI Koichi The Univ.of Tokushima, Dept.of Math.& Natural Sc., Lecturer, 総合科学部, 講師 (90219890)
FUKAGAI Yoshinobu The Univ.of Tokushima, Fac.of Technology, Assosiaite P., 工学部, 助教授 (90175563)
NARUKAWA Kimiaki Naruto Edu.Univ., Fac.of School Ed., Professor, 学校教育学部, 教授 (60116639)
YAMADA Yoshio Waseda Univ.Dept.of Mathematics, Professor, 理工学部, 教授 (20111825)
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| Project Period (FY) |
1997 – 1998
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| Keywords | p-Laplacian / *-Laplacian / limit eigenvalue problem / Poincare inequality / degenerate elliptic equation / reaction-diffusion equations / delay differential equation / blow up |
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| Research Abstract |
1) The p-Laplace operators is well known as the non-linear modification of the usual Laplacian. These operators or their perturbed operators arise in the model equations for the elastic membrane, nonlinear diffusion phenomena and so on. Moreover, the limit state of solutions at p infinity is of great interest from the mathematical or technological view points. The eigenvalue problem of p-Laplacian has been studied by many authors. Since this problem can be dealt with as a variational problem, many results has been known. However, its limit problem at p infinity had been known because it cannot be described in a variatinal problem. We formulate such problem using the notion of the viscosity solution and obtain some results for the limit eigenvalues and the associate eigenfunctions.
2) In the ecological model, a reaction-diffusion equation has the nonlinear diffusion with the p-Laplace operator when the diffusion depends on the population pressure nonlinearly. Yamada has studied such equation and obtain the unique and global existence of a solution and sonic results on the set of stationary solutions. He also study the 3 species cooperative competition-diffusion systems with linear diffusion, and obtain the necessary and sufficient condition to the existence of the coexistence solutions.
3) Murakami has studied the asymptotic behavior of the solution for several higher dimensional delay differential equations and obtain the existence of periodic solutions which are bifurcated from the equilibrium, in particular, the explicit expressions of the bifurcated periodic solutions.
4) Kohda has obtained some conditions on initial value for parabolic problem which guarantee the blow-up of a solution. Moreover, he had shown the behavior of blow-up solution near blow-up time, that is blow-up patterns.
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