Mathematical Sciences: Studies in Nonlinear Analysis

该奖项支持的工作将集中在非线性微分方程解的行为的几个方面。该项目的第一部分将涉及多个空间维度上的多孔介质方程的解。该方程的Graveleau解是由相关的一阶常微分方程组的轨道生成的径向对称方程的解。这种类型的解往往是半径的幂；这种幂通常被称为反常指数。这种现象也出现在许多其他例子中。最近的工作提出了计算指数精确值的方法。对这个值的了解可以衡量解的持有者连续性。第二个研究方向是具有反应的多孔介质流动在高维空间中的稳定化问题。具体地说，将对具有立方强迫项的方程的正解和负解进行分类。有关这些解决办法的信息预计将在对一般情况的研究中发挥重要作用。超导量子干涉装置也在继续开展工作，这种装置在许多应用中被用作灵敏的磁场探测器。它由两个具有恒定偏置电流的耦合约瑟夫森点结组成。动力学用一对常微分方程组来描述。在分析系统时，我们得到一个Hill方程，它有三个简单的本征值，其余的都是双本征值。为了了解解的稳定性，有必要了解双本征值在扰动下的行为。在寻找特征值的替代表达式方面正在进行工作，这可能会对解的行为有更多的解释。

Work supported by this award will focus on several aspects of the behavior of solutions to nonlinear differential equations. The first part of the project will be concerned with solutions of the porous medium equation in more than one space dimension. The Graveleau solution of this equation is a solution of a radially symmetric equation which is generated by an orbit of a related first order system of ordinary differential equations. Solutions of this type tend to a power of the radius; this power is often referred to as the anomalous exponent. The phenomenon occurs in many other examples as well. Recent work suggests methods for calculating the precise value of the exponent. Knowledge of this value measures the Holder continuity of solutions. A second line of research concerns the stabilization in higher space dimensions for porous medium flow with reaction. Specifically, work will be done to classify positive and negative solutions to the equation with a cubic forcing term. Information about these solutions is expected to play an important role in the studies of the general case. Continuing work is also being carried out on superconducting quantum interference devices which are used in many applications as a sensitive detector of magnetic fields. It consists of two coupled Josephson point junctions with a constant bias current. The dynamics are described by a pair of ordinary differential equations. In analyzing the system one is led to a Hill equation with three simple eigenvalues and all the rest double. In order to understand the stability of solutions it is necessary to understand how the double eigenvalues behave under perturbation. Work is being done in seeking alternative expressions for the eigenvalues which may shed more light on solution behavior.