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

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640228/
• 关键词: reaction-diffusion equation; Lotka-Volterra type; positive steady state; bifurcation; stability; p-Laplacian; Lotka-volterra型; 比較定理; Lotka-Volterraモデル; 強最大値原理; 定常解; reaction-diffusion equation; Lotka-Volterra type; positive steady state; bifurcation; stability; p-Laplacian; Lotka-volterra型; 比較定理; Lotka-Volterraモデル; 強最大値原理; 定常解

(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）分析具有交叉扩散术语的反应扩散系统：我们已经讨论了具有交叉扩散和Lotka-Volterra型反应的反应扩散系统。这些系统出现在数学生物学中。从数学上讲，为存在时间全球解决方案的存在，并获取有关阳性固定溶液（生物学上共存状态）的结构的信息非常重要。至于非平稳问题，在一个和两个空间维度中获得了全局存在结果。对于零dirichlet边界条件的固定问题，我们已经研究了正固定溶液的独特性和非独特性以及它们存在的足够条件。事实证明，我们的系统承认有多重的后代解决方案。此外，数值模拟表现出阳性固定溶液的复杂结构，例如来自半主动溶液的对称溶液的分叉化，并在对称溶液中对不对称溶液进行对称溶液的分叉化。（2）分析与p-laplacian和formistic termenty the Cribistian和degracise the demane the Prinace the Prinacy ancy pranenace the prabician cormenace cormenty of praplacian的分析。显着的非线性现象。我们已经对较高空间维度和一个维度的固定解决方案的结构获得了令人满意的理解。特别是，我们还研究了固定溶液的曲线，并证明了源自退化扩散的扁平帽子的有趣结果。此外，我们可以显示有关非平稳解决方案的时间和空间变化的有趣信息。