Asymptotic analysis of ordinary differential equations, and its application to partial differential equations

常微分方程的渐近分析及其在偏微分方程中的应用

基本信息

批准号：
12640179
负责人：
USAMI Hiroyuki
金额：
$ 1.28万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

(1) Quasilinear ODEs of seond-order, which are generalizations of Emden's equation, are considered. Asymptotic representations of positive solutions are obtained explicitly. When nonlinear terms have singularities at the origin, uniqueness of decaying positive solutions is established.(2) Two-term quasilinear ODEs of fourth-order are considered. Neessary and/or sufficient conditions are established for them to have no positive solutions existing near the infinity. Generalizations and applications of these results to 4-dimensional ordinary differential systems are also obtained.(3) As an application of the results in (1), we obtain sufficient conditions for some types of quasilinear exterior elliptic BVPs to have positive solutions with specified asymptotic behavior near the infinity. As an application of the results in (2), we obtain sufficient conditions for some types of semilinear 2-dimensional exterior elliptic problems to have no positive solutions existing near the infinity.(4) Eigenvalue problems for second-order semilinear ODEs on finite intervals are studied. We establish asymptotic properties of (variational) eigenvalues and eigenfunctions. Eigenvalue problems for n-th order linear ODEs are also studied on infinite intervals. We extend well-known Sturmian theory to these problems partially.(5) We consider self-similar solutions of parabolic systems introduced by Keller and Segel to describe aggregation phenomena of molds due to chemotaxis. We find that such solutions must be radially symmetric, and then clarify the relation between parameters and various norms of solutions.

（1）考虑到Emden方程的概括，被列出的准线性OD被考虑。明确获得阳性溶液的渐近表示。当非线性术语的起源具有奇异性时，就会建立衰减阳性溶液的独特性。（2）考虑了第四阶的两项准线性ODE。为他们建立了Neessary和/或足够的条件，使他们没有在无穷大附近存在正面解决方案。这些结果的概括和应用也获得了4维常规差异系统。（3）作为（1）结果的应用，我们为某些类型的准外椭圆形BVP提供了足够的条件，具有在无穷大附近的特定渐进性行为的正面解决方案。作为结果在（2）中的应用，我们为某些类型的半线性二维外部椭圆形问题获得了足够的条件，即在无限大度附近没有存在正面解决方案。（4）在有限间隔内研究了二阶半odes的特征值问题。我们建立了（变化）特征值和特征功能的渐近特性。还以无限的间隔研究了第n阶线性ODE的特征值问题。我们将众所周知的斯特里亚理论扩展到部分问题。（5）我们考虑了凯勒和塞格尔引入的抛物线系统的自相似解决方案，以描述由于趋化性而引起的霉菌的聚集现象。我们发现，这种溶液必须是径向对称的，然后阐明参数与各种溶液规范之间的关系。