Mathematical Sciences: Studies in Nonlinear Analysis

该奖项支持的工作将集中在几个方面 非线性微分方程解的行为。 项目的第一部分将涉及以下问题的解决方案： 多孔介质方程在一个以上的空间维。 的 这个方程的Graveleau解是一个径向 一个对称方程，由一个相关的 一阶常微分方程组 解决方案 这种类型的倾向于半径的幂;这种幂通常是 称为反常指数。 这种现象发生在 还有很多其他的例子。 最近的工作提出了 计算指数的精确值。 知道这起 值度量解的保持器连续性。 第二个研究方向是关于 具有反应的多孔介质流动的更高空间维度。 具体来说，将做好正反分类工作 方程的三次强迫项的解。 信息 关于这些解决方案，预计将发挥重要作用， 一般案例的研究。 超导方面的工作也在继续进行 在许多应用中使用的量子干涉装置 作为磁场的灵敏探测器。 它包括两 耦合约瑟夫森点结与恒定的偏置电流。 动力学由一对常微分方程描述 方程 在分析该系统时，我们导出希尔方程 三个简单的特征值，其余的都是双倍的。 为了 为了理解溶液的稳定性，必须 理解双特征值在微扰下的行为。 目前正在努力寻找替代的表达方式， 特征值，这可能会揭示更多的解决方案的行为。

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.