Qualitative Studies of Parabolic Partial Differential Equations

抛物型偏微分方程的定性研究

• 批准号: 0400702
• 负责人: Peter Polacik
• 金额: $ 9万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400702&HistoricalAwards=false
• 关键词: Qualitative; Studies; Parabolic; Partial; Differential

Proposal DMS-0400702Title: Qualitatitve studies of parabolic partial differential equationsPI: Peter Polacik, University of Minnesota (Twin Cities)ABSTRACTThe abstract follows:The project will develop along the following central themes: symmetry properties of positive solutions of parabolic PDEs, principal Floquet bundles and exponential separation, large time behavior of solutions of parabolic PDEs and the dynamics of quasilinear competition-diffusion systems.Symmetry (radial or reflectional) plays an important role in qualitative analysis of solutions of PDEs. By now, a fairly general understanding of symmetry of positive solutions has been achieved for second order elliptic equations on both bounded and unbounded domains, and for parabolic equations on bounded domains. For nonautonomous parabolic equations on unbounded domains the symmetry problem has not been addressed so far. This problem is more intriguing; standard methods used in earlier symmetry results do not give desired conclusions. Exponential separation, associated with the principal Floquet bundle of linearized parabolic equations, is among new techniques that can facilitate the analysis. The principal Floquet bundle is an interesting topic in its own right. It is a natural extension of the concept of principal eigenfunction of elliptic operators to nonautonomous parabolic operators which has already proved very useful in the study of nonlinear equations. Both its basic properties and applications are to be further investigated. A better understanding is needed to make this tool applicable to problems on unbounded domains, however, ideas related to this concept have already triggered progress in the symmetry problem. The next topic, large-time behavior, includes in particular a basic question on the semilinear heat equation, as to whether all bounded solutions on multidimensional domains converge to an equilibrium. This has been already been answered (negatively) for spatially inhomogeneous equations. For homogeneous equations, this long standing open problem will require new ideas. Competition-diffusion systems that are to be considered in the project have their origins in ecology. Bifurcations of steady states, their stability and as complete as possible an understanding of global dynamics is expected to shed light on coexistence of species (of phenotypes) with different dispersal rates. While a large class of semilinear systems has already been successfully treated, more realistic quasilinear systems call for a new approach.In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of nonlinear evoution equations, such as reaction-diffusion equations. As a rule, such nonlinear equations can "never be solved". Letting aside exact solutions which are very rarely available, approximate methods, even with the presently available computation power, can seldom provide answers to questions of global nature, for example, questions on the behavior of solutions on infinite time intervals or collective behavior of solutions (structural stability). Yet, for the internal development of the theory of PDEs as well as for improvement of their modeling relevance in other sciences, such questions are important to ask and answer. For this purpose, many methods of qualitative analysis of nonlinear PDEs have been developed, mainly in the last 2-3 decades. The present project relies on these methods, combining classical PDE techniques with dynamical systems ideas, and, at the same time, attempts at development of new techniques. Among main objectives of the project is the description of the large time behavior of the solutions. Spatial profiles (symmetries) and temporal behavior (asymptotic periodicity, stabilization to equilibria) will both be examined.

提案DMS-0400702标题：抛物型偏微分方程的定性研究PI：Peter Polacik，明尼苏达大学（双子城）摘要摘要如下：该项目将沿着以下中心主题发展：抛物型偏微分方程正解的对称性，主Floquet丛和指数分离，抛物型偏微分方程解的大时间行为和拟线性竞争扩散系统的动力学。对称性（径向或反射）在偏微分方程解的定性分析中起着重要的作用。 到目前为止，对于有界和无界区域上的二阶椭圆型方程，以及有界区域上的抛物型方程的正解的对称性已经有了相当普遍的认识。对于无界区域上的非自治抛物型方程，对称性问题至今尚未得到解决。 这个问题更有趣;早期对称性结果中使用的标准方法并没有给出预期的结论。指数分离，与线性化抛物方程的主要Floquet丛，是新的技术，可以方便的分析。主Floquet丛本身就是一个有趣的话题。它是椭圆型算子的主本征函数概念到非自治抛物型算子的自然推广，在非线性方程的研究中已被证明是非常有用的。其基本性质和应用有待进一步研究。需要更好的理解，使这个工具适用于无界域的问题，然而，与这个概念相关的想法已经引发了对称性问题的进展。 下一个主题，大时间的行为，包括特别是一个基本问题的半线性热方程，是否所有有界的解决方案多维域收敛到一个平衡。对于空间非齐次方程，这个问题已经得到了（否定的）回答。 对于齐次方程，这个长期存在的开放问题将需要新的想法。在这个项目中要考虑的竞争扩散系统有其生态学的起源。分叉的稳定状态，其稳定性和尽可能完整的全球动态的理解，预计将揭示物种（表型）的共存与不同的扩散率。虽然已经成功地处理了一大类半线性系统，但更现实的拟线性系统需要一种新的方法，用较少的技术术语来说，该项目可以被描述为非线性演化方程（如反应扩散方程）解的定性或几何分析。一般来说，这种非线性方程“永远无法求解”。撇开很少可用的精确解不谈，近似方法，即使有目前可用的计算能力，也很少能回答全局性质的问题，例如，无限时间间隔上的解的行为或解的集体行为（结构稳定性）。然而，对于偏微分方程理论的内部发展以及改进其在其他科学中的建模相关性，这些问题的提出和回答是重要的。为此，许多非线性偏微分方程的定性分析方法已经发展起来，主要是在过去的2- 30年。本项目依赖于这些方法，结合经典偏微分方程技术与动力系统的想法，并在同一时间，在新技术的发展尝试。该项目的主要目标之一是描述解决方案的大时间行为。空间分布（对称性）和时间行为（渐近周期性，稳定到平衡）都将被检查。