Qualitative Studies of Nonlinear Elliptic and Parabolic Equations

非线性椭圆方程和抛物线方程的定性研究

• 批准号: 1565388
• 负责人: Peter Polacik
• 金额: $ 18万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565388&HistoricalAwards=false
• 关键词: Qualitative; Studies; Nonlinear; Elliptic; Parabolic

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.

该项目与非线性抛物线和椭圆形偏微分方程有关。抛物线方程是进化方程 - 未知函数（即解决方案）取决于一个或几个空间变量，而另外一个区别的变量起着时间的作用。这种方程在应用科学的模型中广泛使用，尤其是化学工程，燃烧理论和生态学。鉴于系统的初始状态，问题是要描述其未来状态。从数学上讲，这转化为对解决方案的空间结构（例如均匀性，对称性，集中度）的理解，以及其时间行为，例如实现时间无关的稳态或周期性行为的方法，或更为复杂的行为的可能性。椭圆方程是方程，其解决方案可以视为抛物线方程（以及许多其他类型的进化方程）的时间独立的解决方案或平衡。因此，自然而然地，对椭圆方程的分析是理解抛物线方程动力学的关键基本步骤之一。 对于本项目特别重要的是稳态的对称特性以及某些椭圆方程的整个稳态集合的全局结构。 对本项目要进行的解决方案的定性分析对于偏微分方程数学理论的内部发展以及改善其建模相关性很重要。 严格的分析即使在当前可用于数值分析的高计算能力的情况下，也具有必不可少的作用。它不仅为原本可强大的计算提供了指南和简化，在许多情况下，定性分析是处理有关非线性方程的一般解决方案的唯一方法。该项目的研究将沿着几个主要主题发展。对于真实线上的抛物线方程，首席研究人员将首先分析前线溶液的行为及其传播露台的方法（行进前线的堆叠系统）。然后，他将仔细研究有关局部拓扑的一般解决方案的准佛法。对于整个空间上的多维抛物线问题，要解决的基本问题之一是，有限的解决方案至少像一维方程的解决方案一样，至少沿着一系列时间融合了平衡。其他两个问题涉及liouville型定理，用于非线性抛物线方程的整个解决方案。在其中一个中，首席研究人员建议一种使用liouville定理的方法，以证明传播多维抛物线问题解决方案的露台的方法。在另一个方面，抛物线偏微分方程和liouville定理中的缩放技术用于分析具有奇异性的解决方案。该领域的一个主要问题是确定Liouville定理有效性的最佳指数范围。 在整个空间上的椭圆方程中，其中一个问题涉及除一个变量以外的所有方案的解决方案。首席研究人员旨在建立在Nondecay变量中的Quasiperiodic的解决方案的存在。他还将继续研究他的对称性和椭圆形方程非负解决方案的节点结构，以及各种抛物线问题的阈值解决方案。