Qualitative Studies of Nonlinear Elliptic and Parabolic Equations

这个项目是关于非线性抛物型和椭圆型偏微分方程。抛物方程是演化方程-未知函数（即，解）取决于一个或多个空间变量和一个起时间作用的更显著的变量。这些方程广泛应用于应用科学的模型中，特别是化学工程、燃烧理论和生态学。给定系统的初始状态，问题是描述它的未来状态。在数学上，这转化为对空间结构的理解（例如，均匀性，对称性，浓度）的解决方案在很大的时间，以及它的时间行为，如接近时间无关的稳定状态或周期性行为，或一个更复杂的行为的可能性。椭圆型方程的解可以看作是抛物型方程（以及许多其他类型的演化方程）的时间无关解或平衡。因此，自然地，椭圆方程的分析是理解抛物方程动力学的关键基本步骤之一。 对本项目特别重要的是某些椭圆方程定态的对称性和整组定态的全局结构。 本项目中进行的解的定性分析对于偏微分方程数学理论的内部发展以及改进其建模相关性非常重要。 严格的分析保持其不可或缺的作用，即使在存在的高计算能力，目前可用于数值分析。它不仅提供了指导方针和简化，否则艰巨的计算，在许多情况下，定性分析是唯一的方法来处理困难的问题，一般解决非线性方程。本项目的研究将沿着沿着几个主要课题展开。对于真实的线上的抛物方程，主要研究者将首先分析类前沿解的行为及其传播阶地（旅行前沿的堆叠系统）的方法。然后，他将仔细研究拟收敛性质的一般解决方案，就一个本地化的拓扑结构。对于整个空间上的多维抛物问题，要解决的基本问题之一是有界解是否收敛到平衡，至少沿着一个时间序列，作为解决方案的一维和二维方程。另外两个问题是关于非线性抛物型方程整体解的刘维尔型定理。在其中之一，主要研究者提出了一种方法，使用刘维尔定理证明的方法来传播梯田的解决方案的多维抛物问题。在另一个中，抛物型偏微分方程的尺度技术和刘维尔定理被用于分析具有奇异性的解决方案。在这方面的一个主要问题是确定指数的有效性刘维尔定理的最佳范围。 在整个空间上的椭圆方程中，其中一个问题涉及到除一个变量外所有变量都衰减为零的解。主要研究人员试图建立的解决方案，是在nondecay变量的准周期的存在。他还将继续致力于他的项目的对称性和节点结构的非负解的椭圆和抛物方程和阈值解决方案在各种抛物问题。

This project is concerned with nonlinear parabolic and elliptic partial differential equations. Parabolic equations are evolution equations--the unknown function (i.e., the solution) depends on one or several spatial variables and one more distinguished variable playing the role of time. Such equations are widely used in models in applied sciences, in particular, in chemical engineering, combustion theory, and ecology. Given an initial state of the system, the problem is to describe its future states. Mathematically, this translates to an understanding of the spatial structure (e.g., homogeneity, symmetry, concentration) of the solution at large times, as well as of its temporal behavior, such as approach to a time-independent steady state or periodic behavior, or possibilities of an even more complicated behavior. Elliptic equations are equations whose solutions can be viewed as time-independent solutions, or equilibria, of parabolic equations (and many other types of evolution equations). Naturally, therefore, analysis of elliptic equations is one of the key basic steps toward understanding the dynamics of parabolic equations. Of particular significance to the present project are symmetry properties of steady states and the global structure of the whole set of steady states for certain elliptic equations. Qualitative analysis of solutions to be carried out in this project is important for the internal development of the mathematical theory of partial differential equations as well as for improvement of their modeling relevance. Rigorous analysis maintains its indispensable role even in the presence of the high computing power currently available for numerical analysis. Not only does it provide guidelines for and simplifications of otherwise formidable computations, in many situations qualitative analysis is the only way to deal with difficult problems concerning general solutions of nonlinear equations. The research in this project will develop along several main topics. For parabolic equations on the real line, the principal investigator will first analyze the behavior of front-like solutions and their approach to propagating terraces (stacked systems of traveling fronts). He will then take a closer look at quasiconvergence properties of general solutions with respect to a localized topology. For multidimensional parabolic problems on the entire space, one of the basic questions to be addressed is whether bounded solutions converge to equilibrium, at least along a sequence of times, as solutions of the one- and two-dimensional equations do. Two other problems deal with Liouville-type theorems for entire solutions of nonlinear parabolic equations. In one of them, the principal investigator suggests a way of using a Liouville theorem in a proof of the approach to propagating terraces for solutions of multidimensional parabolic problems. In the other one, scaling techniques in parabolic partial differential equations and a Liouville theorem are used for analyzing solutions with singularities. A major problem in this area is to determine the optimal range of exponents for the validity of the Liouville theorem. In elliptic equations on the entire space, one of the problems concerns solutions that decay to zero in all but one variable. The principal investigator seeks to establish the existence of solutions that are quasiperiodic in the nondecay variable. He will also continue working on his projects on symmetry and the nodal structure of nonnegative solutions of elliptic and parabolic equations and on threshold solutions in various parabolic problems.