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

(1)考虑了二阶拟线性常微分方程组，它是Emden方程的推广。明确地得到了正解的渐近表示。当非线性项在原点有奇性时，建立了衰减正解的唯一性。(2)考虑了二项四阶拟线性常微分方程组。建立了它们在无穷远附近不存在正解的必要条件和/或充分条件。(3)作为文(1)中结果的应用，我们得到了某些类型的拟线性外椭圆边值问题在无穷远附近有正解的充分条件。作为(2)中结果的应用，我们得到了某些类型的半线性二维椭圆型外问题在无穷远附近不存在正解的充分条件。(4)研究了有限区间上二阶半线性常微分方程组的特征值问题。我们建立了(变分)特征值和特征函数的渐近性质。还研究了无穷区间上n阶线性常微分方程组的特征值问题。我们将著名的Sturmian理论部分推广到这些问题。(5)考虑Keller和Segel提出的抛物型系统的自相似解来描述由于趋化性而导致的霉菌聚集现象。我们发现这类解一定是径向对称的，并阐明了参数与解的各种范数之间的关系。

(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.