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）反应数学生物学中具有非线性扩散的扩散系统和（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下式给出具有拟线性扩散的微分方程组，其形式为 mu_t=Δ[φ(u,v)u]+au(1-u-v)， v_t=Δ[ψ(u,v)v]+bv(1+du-v)，在均匀狄利克雷边界条件下，其中u和v分别表示猎物和捕食者物种的种群密度，众所周知，相应的。平稳问题在适当的条件下具有正稳态。我们的主要兴趣是在 φ(u,v)=1 和 4,φ(u,v) 的情况下得出有关每个正稳态的轮廓和稳定性的有用信息。 =1+β u，我们已经证明，如果 β 足够大并且施加一些其他条件，那么平稳问题至少有三个正解。此外，还研究了每个正解的稳定性。第二个问题由 u_t 给出。 =ε^2u_<xx>+u(1-u)(u-a(x)) 具有齐次诺伊曼边界条件，其中 0<a(x)<1 当 ε 足够小时，已知该问题存在各种情况。稳态的种类特别是，我们对具有过渡层和尖峰的稳态感兴趣。这里，解决方案的过渡层是指 u(x) 的一部分，其中 u(x) 在很短的时间内从 0 到 1 或 1 到 0。 Ai-Chen-Hastings 和我们的小组独立研究了这种振荡解，并证明任何过渡层仅出现在 x 的邻域中，使得 a(x)=1/2 并且出现任何尖峰。仅在一个邻域内x 使得 a(x) 取其局部最大值或最小值。我们还建立了有关多过渡层和多尖峰的轮廓、它们的位置以及过渡层稳态解的轮廓和稳定性之间的关系的更多信息。较少的