Mathematical Sciences: Studies in Nonlinear Analysis

****************************************************************************** 非线性分析研究 拟议的研究是针对两种类型的问题 关于非线性微分方程解的性质 方程 这些问题中的第一个涉及到存在和 多孔介质自相似解的中间渐近性 介质方程（退化的非线性抛物方程）和 气体动力学的纳维-斯托克斯方程 这里的目标是获得 流动中各种临界转变的渐近描述 所涉及的，如在多孔介质流孔（聚焦）的关闭 或者气体动力学中的冲击波内爆。 第二个问题 研究了约瑟夫森负载耦合天线阵的动态特性 点连接。这些阵列由非线性系统建模， 常微分方程，研究的目的是 了解稳定和不稳定的各种开始， 锁相解决方案 将特别强调 预测锁相失稳和动态 这种损失的后果。 偏微分方程的自相似解是 一个特别简单的结构的解决方案， 方程的对称性。 它们经常出现在渐近 描述流程中的关键转换，因为这些 转换是方程本身的特征， 不依赖于特定流的细节。 这方面的一个例子 在工程实践中出现了与涂料流动有关的现象 非常粘稠的液体。 在这种流动中填充孔的过程是 通常由多孔介质方程描述，我们的结果将是 在发展数值方法的研究和控制这种有用的 流程. 负载耦合约瑟夫森结的大阵列用于 诸如微波发生器和参量发生器之类的许多电子设备 放大器. 在所有这些设备中，最理想的操作条件 是所有结都一致振荡的一种。 不幸的是，对于各种结特性和负载， 这种相位锁定状态的配置不稳定。因此 重要的是要知道如何预测的稳定性和不稳定性， 相位锁定状态的参数值为任何特定的阵列。

****************************************************************************** STUDIES IN NONLINEAR ANAYSIS: ABSTRACT The proposed research is directed towards two types of problems concerning the behavior of solutions to nonlinear differential equations. The first of these problems concerns the existence and intermediate asymptotic behavior of self-similar solutions to the porous medium equation (a degenerate nonlinear parabolic equation) and to the Navier-Stokes equations of gas dynamics. The objective here is to obtain the asymptotic description of various critical transitions in the flows involved, such as the closing of holes (focusing) in porous medium flow or the implosion of shock waves in gas dynamics. The second problem concerns the study of the dynamics of of load coupled arrays of Josephson point junctions. These arrays are modeled by nonlinear systems of ordinary differential equations, and the object of the study is to understand the onset of stability and instability of various kinds of phase locked solutions. Particular emphasis will be placed on predicting the loss of stability of phase lock and the dynamical consequences of such a loss. Self-similar solutions to partial differential equations are solutions of a particularly simple structure determined by the underlying symmetries of the equation. They often arise in the asymptotic description of critical transitions in the flow, because these transitions are characteristic of the equations themselves and do not depend on the details of the particular flow. An example of this phenomenon occurs in engineering practice in connection with coating flows of very viscous fluids. The process of filling holes in such flows is often described by a porous medium equation, and our results will be useful in developing numerical methods for the study and control of such processes. Large arrays of load coupled Josephson junctions are used in many electronic devices such as microwave generators and parametric amplifiers. In all these devices the most desirable operating condition is one in which all of the junctions oscillate in unison. Unfortunately, for various junction characteristics and load configurations this phase locked state is not stable. It is therefore important to know how to predict the stability and instability of the phase locked state in terms of the parameter values for any particular array.