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Study of nonlinear prabolic systems and related elliptic systems

非线性抛物线系统及相关椭圆系统的研究

基本信息

批准号：
09640228
负责人：
YAMADA Yoshio
金额：
$ 2.24万
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：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

(1) Analysis of reaction diffusion systems with cross-diffusion terms : We have discussed reaction diffusion systems with cross-diffusion and reaction of Lotka-Volterra type. These systems appear in mathematical biology. Mathematically, it is very important to derive sufficient conditions for the existence of time-global solutions and get information on the structure of positive stationary solutions (biologically, coexistence states). As to the non-stationary problem a global existence result has been obtained in one and two space-dimensions. For the stationary problem with zero Dirichlet boundary condition, we have studied uniqueness and non-uniqueness of positive stationary solutions as well as sufficient conditions for their existence. It is proved that our system admits multiple existence of postive solutions. Moreover, numerical simulations exhibit complicate structure of positive stationary solutions such as bifurcation of symmetric solutions from semitrivial solutions and, additionally, bifurcation of asymmetric solutions from symmetric ones.(2) Analysis of quasilinear parabolic equations with p-Laplacian and logistic terms : Although the nonlinearity and degeneracy of p-Laplacian brings about the difficulty, it also gives remarkable nonlinear phenomenon. We have obtained satisfactory understanding on the structure of stationary solutions in higher space dimension as well as one dimension. In particular, we also have studied profiles of stationary solutions and proved interesting results on flat hats which stem from degenerate diffusion. Furthermore, we could show interesting information on the temporal and spatial change of non-stationary solutions.

(1) 具有交叉扩散项的反应扩散系统的分析：我们讨论了具有交叉扩散项的反应扩散系统和Lotka-Volterra 类型的反应。这些系统出现在数学生物学中。在数学上，推导时间全局解存在的充分条件并获得正稳态解（生物学上的共存状态）结构的信息非常重要。对于非平稳问题，在一维和二维空间上得到了全局存在结果。对于零狄利克雷边界条件的平稳问题，我们研究了正平稳解的唯一性和非唯一性及其存在的充分条件。证明了我们的系统承认正解的多重存在。此外，数值模拟表现出正稳态解的复杂结构，例如对称解与半平凡解的分叉，以及非对称解与对称解的分叉。(2)带有p-拉普拉斯和逻辑项的拟线性抛物型方程的分析：虽然p-拉普拉斯的非线性和简并性带来了困难，但也给出了显着的非线性现象。我们对高维空间和一维的平稳解的结构已经取得了令人满意的理解。特别是，我们还研究了平稳解的轮廓，并证明了源自简并扩散的平帽的有趣结果。此外，我们可以显示有关非平稳解的时间和空间变化的有趣信息。