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

在该项目中，我们研究了以下两种方程式的解决方案的结构：（a）数学生物学非线性差异的反应扩散系统和（b）半线性差异方程式描述了数学生物学中的第一个问题的第一个问题是由具有差异方程的系统差异的系统差异给出的。在同质的dirichlet边界条件下，Formula_T =Δ[φ（U，V）U]+AU（1-U-V），V_T =δ[ψ（U，V）V]+BV（1+DU-V），在均质的Dirichlet边界条件下。在这里，u和v分别表示猎物和捕食者物种的人口密度。众所周知，相应的固定问题在适当的条件下具有正稳态。我们的主要兴趣是获取有关每个正稳态的轮廓和稳定性的有用信息。在φ（u，v）= 1和4的情况下，φ（u，v = 1+βu，我们已经表明，如果β足够大，并且提出了其他一些条件，则固定问题至少具有三个积极的解决方案。此外，稳定性或在…中的稳定性或更高的稳定性也更高的稳定性也被研究了第二个问题。还研究了第二个问题。 Neumann边界条件，其中0 <a（x）<1时，尤其是该问题尤其是稳态的解决方案。 A-Chen-Hastings和我们的小组独立证明，任何过渡层仅在X的邻居中出现，因此A（X）= 1/2，并且任何Spike仅在X附近出现，以使A（X）占据其局部最大值或最小值。我们还建立了更多有关多转移层和多尖峰的配置文件，它们的位置以及稳态解决方案与过渡层之间的稳定性之间的关系。较少的