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Qualitative Properties of Solutions of Nonlinear Elliptic and Parabolic Equations

非线性椭圆方程和抛物方程解的定性性质

基本信息

批准号：
1856491
负责人：
Peter Polacik
金额：
$ 29.03万
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：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856491&HistoricalAwards=false
关键词：
Qualitative Properties Solutions Nonlinear Elliptic

项目摘要

This project is concerned with nonlinear parabolic and elliptic partial differential equations (PDEs), with special focus on problems posed on an entire Euclidean space. Parabolic equations are evolution equations?the unknown function, or 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. Given an initial state of the system, the main goal is to describe its future states. Mathematically, this translates to questions about a possible development of singularities in the solutions, and, in the absence of such singularities, about the behavior of the solutions as time increases to infinity. One asks if the solution approaches in some way a time-independent steady state or if it may exhibit a more complicated behavior. Elliptic equations on Euclidean spaces represent steady states (equilibria), solitary waves, traveling fronts, or self-similar solutions of many different types of evolution PDEs. Naturally, therefore, analysis of elliptic equations is one of the key basic steps toward understanding of the dynamics of these evolution equations. The project?s problems in elliptic equations concern qualitative properties, such as symmetry, periodicity, and more complex oscillatory behavior of individual solutions, as well as the global structure of the solutions, such as their multiplicity and bifurcations (changes as parameters in the equation vary). Qualitative analysis of solutions to be carried out in this project is important for the internal development of the mathematical theory of PDEs as well as for improvement of their modeling relevance. Although the project is mainly theorical, its results concerning fundamental properties of solutions could be of interest in research fields beyond PDEs. For example, even with high computing power currently available for numerical analysis, computations involving nonlinear PDEs are often formidable without a guideline from qualitative analysis. Also, when a specific PDE model from applied science is to be investigated, general qualitative results on possible behavior of solutions of equations of the given type provide a valuable information. This project contains component projects and activities for graduate students, and the award provides graduate student research assistantship and summer support, and support for student participation at conferences.The research in this project will develop along several main topics. In certain elliptic equations on the entire space, one of the problems concerns solutions which decay to zero in all but one variable. Employing techniques from the center manifold and KAM theories, the PI wants to examine the existence of solutions which are quasiperiodic in the non-decay variable. Another class of elliptic equations to be considered arises as an equation for self-similar solutions of the semilinear heat equation. The PI will study the multiplicity of the solutions and their bifurcations from a singular solution as the exponent in the power nonlinearity varies. For parabolic equations on the real line, the PI will continue his research on quasiconvergence properties of solutions with respect to a localized topology. For multidimensional semilinear parabolic equations on the entire space, one of the basic questions to be addressed is whether in high enough spatial dimensions, the solutions may exhibit some sort of oscillatory behavior while staying away from steady states on any sufficiently large bounded region. Two other problems deal with Liouville-type and classification theorems for entire solutions (that is, solutions defined for all times, positive and negative) of nonlinear parabolic equations. In one of them, the goal is to prove the nonexistence of positive entire solutions for an optimal Sobolev-subcritical range of exponents. In another one, the existence of nonstationary entire solutions is to be investigated for a range of supercritical exponents. Liouville and classification theorems have many interesting applications in the theory of blowup of solutions of parabolic equations and some of them, such as the characterization the type of blowup and existence of spatial blowup profiles, also belong to the expected outcome of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目关注的是非线性抛物和椭圆偏微分方程（PDE），特别关注整个欧几里德空间上的问题。抛物方程是演化方程吗？未知函数或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。这类方程在应用科学的模型中被广泛使用。给定系统的初始状态，主要目标是描述其未来状态。在数学上，这转化为关于解中奇点的可能发展的问题，以及在没有这种奇点的情况下，关于解随着时间增加到无穷大的行为的问题。有人问，如果解决方案以某种方式接近时间无关的稳定状态，或者如果它可能表现出更复杂的行为。欧几里得空间上的椭圆方程代表许多不同类型的演化偏微分方程的稳态（平衡）、孤立波、行进前沿或自相似解。因此，自然地，椭圆方程的分析是理解这些演化方程动力学的关键基本步骤之一。项目？椭圆型方程的问题涉及定性性质，如对称性，周期性，和更复杂的振荡行为的个别解决方案，以及整体结构的解决方案，如他们的多重性和分叉（变化的参数在方程中的变化）。在这个项目中进行的解决方案的定性分析是很重要的内部发展的数学理论的偏微分方程，以及为改善其建模的相关性。虽然该项目主要是理论性的，但其关于解决方案的基本性质的结果可能在偏微分方程以外的研究领域引起兴趣。例如，即使目前有很高的计算能力可用于数值分析，涉及非线性偏微分方程的计算往往是可怕的，没有从定性分析的指导。此外，当一个特定的偏微分方程模型从应用科学是要调查，一般定性结果的可能行为的方程的解决方案的给定类型提供了一个有价值的信息。该项目包含研究生的组成项目和活动，该奖项提供研究生研究助理和夏季支持，并支持学生参加会议。该项目的研究将沿着沿着几个主要主题发展。在整个空间上的某些椭圆方程中，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。利用中心流形和KAM理论的技术，PI想要检查在非衰减变量中准周期解的存在性。另一类椭圆型方程被认为是一个方程的自相似解的半线性热方程。 PI将研究的多重性的解决方案和他们的分歧，从一个奇异的解决方案的指数在幂非线性变化。对于真实的直线上的抛物方程，PI将继续研究局部拓扑解的拟收敛性质。对于整个空间上的多维半线性抛物方程，需要解决的基本问题之一是，在足够高的空间维度下，解是否可能在任何足够大的有界区域上表现出某种振动行为，同时远离稳态。另外两个问题涉及非线性抛物方程的整体解（即定义为所有时间的正解和负解）的刘维尔型和分类定理。在其中之一，我们的目标是证明不存在的正整体解的最佳Sobolev次临界范围的指数。在另一个问题中，研究了在一定范围的超临界指数下非平稳整体解的存在性。Liouville定理和分类定理在抛物型方程解的爆破理论中有许多有趣的应用，其中一些应用，如爆破类型的表征和空间爆破轮廓的存在性，这一奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查进行评估，被认为值得支持的搜索.