2005 Fiscal Year Final Research Report Summary
Research of System of Nonlinear Diffusion Equations and Related Elliptic Differential Equations
| Project/Area Number |
15540216
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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 |
Global analysis
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| Research Institution | Waseda University |
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Principal Investigator |
YAMADA Yoshio Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20111825)
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| Co-Investigator(Kenkyū-buntansha) |
OTANI Mitsuharu Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (30119656)
TANAKA Kazunaga Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (20188288)
NAKASHIMA Kimie Tokyo University of Marine Science and Technology, Faculty of Marine Science, Associate Professor, 海洋科学部, 助教授 (10318800)
TAKEUCHI Shingo Kogakuin University, Faculty of Engineering, Lecturer, 工学部, 講師 (00333021)
KUTO Kousuke Fukuoka Institute of Technology, Faculty of Engineering, Lecturer, 工学部, 講師 (40386602)
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| Project Period (FY) |
2003 – 2005
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| Keywords | reaction-diffusion / quasilinear diffusion / internal transition layer / spike / steady-state solution / stability / prey-predator model / Morse index |
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| Research Abstract |
In this project, we have studied the structure of solutions for the following two types of equations : (a) reaction diffusion systems with nonlinear diffusion in mathematical biology and (b) semilinear diffusion equations describing phase transition phenomena
The first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the form
u_t=Δ[φ(u,v)u]+au(1-u-v), v_t=Δ[ψ(u,v)v]+bv(1+du-v),
under homogeneous Dirichlet boundary conditions. Here u and v denote population densities of prey and predator species, respectively. It is well known that the corresponding stationary problem has a positive steady-state under a suitable condition. Our main interest is to derive useful information on profile and stability of each positive steady-state. In case φ(u,v)=1 and 4,φ(u,v=1+β u, we have shown that the stationary problem has at least three positive solutions if β is sufficiently large and some other conditions are imposed. Moreover, stability or in
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stability of each positive solution is also investigated.
The second problem is given by u_t=ε^2u_<xx>+u(1-u)(u-a(x)) with homogeneous Neumann boundary condition, where 0<a(x)<1. When ε is sufficiently small, it is known that this problem admits various kinds of steady-state solutions. In particular, we are interested in steady state with transition layers and spikes. Here transition layer for a solution means a part of u(x) where u(x) drastically changes from 0 to 1 or 1 to 0 in a very short interval. Such oscillating solutions have been studied by Ai-Chen-Hastings and our group, independently. It has been proved that any transition layer appears only in a neighborhood of x such that a(x)=1/2 and that any spike appears only in a neighborhood of x such that a(x) takes its local maximum or minimum. We have also established more information on profiles of multi-transition layers and multi-spikes, their location and the relationship between profile and stability of steady-state solution with transition layers. Less
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