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Research of System of Nonlinear Diffusion Equations and Related Elliptic Differential Equations

非线性扩散方程组及相关椭圆微分方程组的研究

基本信息

批准号：
15540216
负责人：
YAMADA Yoshio
金额：
$ 2.18万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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 phenomenaThe first problem in mathematical biology is given by a system of differential equations with quasilinear diffusion of the formu_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 … More 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

在这个项目中，我们研究了以下两类方程的解的结构：（a）数学生物学中具有非线性扩散的反应扩散系统和（B）描述相变现象的半线性扩散方程数学生物学中的第一个问题由具有拟线性扩散的微分方程组给出，方程组为：u_t=Δ[φ（u，v）u]+Au（1-u-v），v_t=Δ[φ（u，v）u]，v）v]+bv（1+du-v）.这里u和v分别表示猎物和捕食者物种的种群密度。众所周知，在适当的条件下，相应的平稳问题具有正平衡态。我们的主要兴趣是获得有用的信息，每个积极的稳态的配置文件和稳定性。当φ（u，v）=1和4，φ（u，v）=1+β u时，如果β充分大，并附加其它条件，我们证明了该平稳问题至少有三个正解.此外，稳定性或 ...更多信息第二个问题为u_t=ε^2u_<xx>+u（1-u）（u-a（x）），其中0<a（x）<1.当ε足够小时，已知该问题存在各种稳态解。特别是，我们感兴趣的过渡层和尖峰的稳定状态。这里，解的过渡层是指u（x）的一部分，其中u（x）在很短的时间间隔内从0到1或从1到0急剧变化。Ai-Chen-Hastings和我们的小组已经独立地研究了这种振荡解。已经证明，任何过渡层只出现在x的邻域中，使得a（x）=1/2，并且任何尖峰只出现在x的邻域中，使得a（x）取其局部最大值或最小值。我们还建立了更多的信息，多过渡层和多尖峰的轮廓，它们的位置和轮廓之间的关系和稳定性的稳态解的过渡层。少