Qualitative Studies of Parabolic Partial Differential Equations

题目：抛物型偏微分方程的定性研究[j]: Peter Polacik，明尼苏达大学（双子城）摘要：本课题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束和指数分离，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（径向对称或反射对称）在微分方程解的定性分析中起着重要作用。目前，对于二阶椭圆型方程在有界域和无界域上的正解对称性，以及抛物型方程在有界域上的正解对称性，已经有了较为普遍的认识。对于无界域上的非自治抛物方程，其对称性问题迄今尚未得到解决。这个问题更有趣；在以前的对称结果中使用的标准方法不能给出期望的结论。与线性化抛物方程的主Floquet束相关联的指数分离是便于分析的新技术之一。主要的Floquet包本身就是一个有趣的话题。它是椭圆算子的主特征函数概念对非自治抛物算子的自然推广，在非线性方程的研究中已被证明是非常有用的。其基本性质和应用还有待进一步研究。要使这个工具适用于无界域上的问题，还需要更好的理解，然而，与这个概念相关的想法已经引发了对称问题的进展。下一个主题，大时间行为，特别包括一个关于半线性热方程的基本问题，即是否所有在多维域上的有界解收敛于平衡态。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，这个长期悬而未决的问题将需要新的想法。项目中要考虑的竞争-扩散系统起源于生态学。稳定状态的分岔及其稳定性以及对全球动力学的尽可能完整的理解有望阐明具有不同扩散速率的物种（表型）的共存。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。在不太技术性的术语中，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。通常，这样的非线性方程“永远无法解”。撇开很少能得到的精确解不谈，近似方法，即使以目前可用的计算能力，也很少能提供全局性质问题的答案，例如，关于解在无限时间间隔上的行为或解的集体行为（结构稳定性）的问题。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而言，这些问题的提出和回答是重要的。为此，主要是在过去的二、三十年中发展了许多非线性偏微分方程的定性分析方法。目前的项目依赖于这些方法，将经典的PDE技术与动力系统思想相结合，同时尝试开发新技术。项目的主要目标之一是描述解决方案的大时间行为。空间剖面（对称性）和时间行为（渐近周期性，稳定平衡）都将被检查。

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.