2004 Fiscal Year Final Research Report Summary
Asymptotic behavior of an aggregating pattern of the reaction diffusion equation with the advection term
| Project/Area Number |
15540128
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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 | University of Miyazaki |
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Principal Investigator |
TSUJIKAWA Tohru University of Miyazaki, Faculty of Engineering, Professor, 工学部, 教授 (10258288)
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| Co-Investigator(Kenkyū-buntansha) |
SENBA Takashi University of Miyazaki, Faculty of Engineering, Professor, 工学部, 教授 (30196985)
KABEYA Yoshitugu University of Miyazaki, Faculty of Engineering, Associate Professor, 工学部, 助教授 (70252757)
YAGI Atsushi Osaka University, Faculty of Engineering, Professor, 大学院・工学研究科, 教授 (70116119)
NAKAKI Tatsuyuki Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (50172284)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Exponential attractor / Chemotaxis model / Singular limit Analysis / Squeezing property |
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| Research Abstract |
1.For the chemotaxis growth model, we show the traveling wave solution with a triple junction in the stripe domain by the numerical simulations. In order to show the existence of the solution, we first construct the approximate solution of this solution by the interface equation corresponding to the original model equation. Moreover, it can be shown die relation of the velocity of the traveling solution and the intensity of the chemotaxis.
2.For the adsorbate-induced phase transition model, we show the existence of the nonnegative global solution and exponential attractor under the periodic boundary condition in two dimensional bounded domain. Due to the appropriate choice of the functional space, we prove that the sqeezing property is hold for the dynamical system obtained from the equation.
3.For the adsorbate-induced phase transition model in the plane, we show the existence of the nonnegative global solutions under the boundary condition such that the solution tends to the constant equilibrium solution at the edge of the plane. In this situation, we can not prove the existence of the exponential attractor.
4.For the adsorbate-induced phase transition model in the bounded domain of the plane, we prove the existence of the global solutions and exponential attractor under the Newman boundary condition. By the numerical simulations, we show the hexagonal and stripe patterns and so on for the parameters in the neighborhood of the bifurcation point of constant equilibrium solution.
5.For the chemotaxis growth model, we need consider the sensitive function with the singularity at the origin from the biological view points. For the bounded domain in the plane, we prove that the solution tends to the trivial solution if the initial data is small with respect to some functional norm. Moreover, there is a nonempty omega limit set in another case.
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