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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型反应。这些系统出现在数学生物学中。在数学上，它是非常重要的，以获得充分条件的存在性的时间整体解决方案，并获得信息的结构上的正稳定的解决方案（生物，共存状态）。对于非定常问题，在一维和二维空间中得到了整体解的存在性。对于具有零Dirichlet边界条件的平稳问题，研究了正平稳解的唯一性和非唯一性以及正平稳解存在的充分条件。证明了该系统存在多个正解。此外，数值模拟还显示了正定态解的复杂结构，如对称解从半平凡解的分叉，以及非对称解从对称解的分叉. (2)含p-Laplacian和logistic项的拟线性抛物方程的分析：虽然p-Laplacian的非线性和退化性给分析带来了困难，但它也给出了显著的非线性现象。我们对高维空间和一维空间中定态解的结构都有了满意的认识。特别是，我们还研究了固定的解决方案，并证明了有趣的结果，源于退化扩散的平顶帽。此外，我们可以显示有趣的信息的时间和空间变化的非平稳的解决方案。