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.

本课题涉及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）取决于一个或几个空间变量和一个扮演时间角色的另一个杰出变量。这些方程被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，问题是描述其未来状态。在数学上，这转化为对大时间解的空间结构（例如，均匀性，对称性，浓度）的理解，以及它的时间行为，例如接近于时间无关的稳态或周期行为，或更复杂行为的可能性。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点。因此，对椭圆方程的分析自然是理解抛物方程动力学的关键基本步骤之一。对本课题特别有意义的是稳态的对称性和某些椭圆方程的整组稳态的整体结构。在这个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对其建模相关性的改进是重要的。即使在目前可用于数值分析的高计算能力存在的情况下，严格分析仍然保持其不可或缺的作用。定性分析不仅为复杂的计算提供指导和简化，而且在许多情况下，定性分析是处理非线性方程通解难题的唯一方法。这个项目的研究将沿着几个主要主题发展。对于实线上的抛物方程，首席研究员将首先分析锋面解的行为及其传播梯田（行进锋面的堆叠系统）的方法。然后，他将进一步研究关于局域拓扑的一般解的拟收敛性质。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界解是否收敛于平衡，至少沿着一系列时间，就像一维和二维方程的解一样。另外两个问题涉及非线性抛物方程全解的liouville型定理。在其中一篇文章中，首席研究员提出了一种使用刘维尔定理来证明多维抛物线问题解的传播梯田方法的方法。另一种是利用抛物型偏微分方程的标度技术和刘维尔定理来分析具有奇异性的解。该领域的一个主要问题是确定刘维尔定理有效性的最佳指数范围。在整个空间上的椭圆方程中，其中一个问题涉及解在除一个变量外的所有变量中衰减为零。主要研究者试图在非衰变变量中建立准周期解的存在性。他还将继续研究椭圆型和抛物型方程非负解的对称性和节点结构，以及各种抛物型问题的阈值解。