Qualitative studies of solutions of nonlinear elliptic and parabolic equations

非线性椭圆方程和抛物方程解的定性研究

• 批准号: 1161923
• 负责人: Peter Polacik
• 金额: $ 19.8万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161923&HistoricalAwards=false
• 关键词: Qualitative; studies; solutions; nonlinear; elliptic

The project is devoted to qualitative studies of partial differential equations (PDE). Building on recent developments, the principal investigator will study symmetry properties and the nodal structure of nonnegative solutions of elliptic PDE. The main goals are to classify spatial domains on which solutions with nontrivial nodal sets can exist and to determine whether such solutions can exist at all for spatially homogeneous equations. In parabolic equations, a tendency of positive solutions to "improve their symmetry" as time increases to infinity is a remarkable example of how parabolic flows can reduce spatial complexity. The principal investigator will continue his study of this interesting asymptotic symmetry phenomenon, while bearing in mind applications of asymptotic symmetry theorems in convergence results for parabolic equations. Other methods will also be employed to address several long-standing problems concerning the convergence of solutions of parabolic PDE to equilibria. Another topic in this project concerns positive solutions of elliptic PDE on the whole Euclidean space that decay to zero in some variables but do not decay in other variables. The symmetry of the solutions with respect to the decay variables and their behavior with respect to the remaining variables will be examined. The principal investigator will also continue his research concerning Liouville-type theorems on the nonexistence of nontrivial solutions for specific classes of nonlinear equations. Scaling techniques based on Liouville theorems have a wide range of applications in the theory of parabolic PDE, which will be further explored in the project. Results of the above projects will be applied in studies of threshold solutions in various parabolic problems. Such solutions occur as separatrices between solutions exhibiting two different kinds of behavior, such as the decay to zero and blow-up in finite time. They have been studied for purely theoretical reasons as well as in connection with quenching and propagation phenomena in applied sciences.In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of nonlinear partial differential equations. Such equations are widely used in models in the applied sciences, in particular, chemical engineering, combustion theory, and ecology. Understanding qualitative properties of solutions is important for the internal development of the mathematical theory of partial differential equations as well as for the improvement of their modeling relevance. For the interpretation of models involving nonlinear partial differential equations, rigorous analysis maintains its indispensable role even in presence of the high computing power currently available for numerical analysis. Not only does it provide guidelines for and simplifications of otherwise formidable computations, but in many situations qualitative analysis is the only way to deal with difficult problems concerning general solutions of nonlinear equations. The present project addresses questions that concern geometric properties of solutions (such as their symmetries when viewed as functions of spatial variables) as well as their behavior with respect to time (periodicity properties, stabilization to equilibria, so-called blow-up in finite time). Development of new mathematical techniques for addressing such questions is an integral part of the project.

该项目致力于偏微分方程（PDE）的定性研究。在最新进展的基础上，首席研究员将研究椭圆偏微分方程非负解的对称性和节点结构。主要目标是对可以存在具有非平凡节点集的解的空间域进行分类，并确定对于空间齐次方程是否可以存在这样的解。在抛物线方程中，随着时间增加到无穷大，正解有“改善对称性”的趋势，这是抛物线流如何降低空间复杂性的一个显着例子。首席研究员将继续研究这种有趣的渐近对称现象，同时牢记渐近对称定理在抛物线方程收敛结果中的应用。还将采用其他方法来解决有关抛物线偏微分方程解收敛到平衡点的几个长期存在的问题。该项目的另一个主题涉及整个欧几里德空间上的椭圆偏微分方程的正解，该解在某些变量中衰减到零，但在其他变量中不衰减。将检查解相对于衰减变量的对称性及其相对于其余变量的行为。首席研究员还将继续研究有关特定类别非线性方程不存在非平凡解的刘维尔型定理。基于刘维尔定理的缩放技术在抛物线偏微分方程理论中具有广泛的应用，该项目将进一步探讨这一点。上述项目的成果将应用于各种抛物线问题的阈值解的研究。此类解作为表现出两种不同行为的解之间的分离而出现，例如在有限时间内衰减到零和爆炸。人们对它们进行了纯粹的理论原因以及与应用科学中的猝灭和传播现象相关的研究。用较少的技术术语来说，该项目可以描述为非线性偏微分方程解的定性或几何分析。此类方程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。了解解的定性性质对于偏微分方程数学理论的内部发展及其建模相关性的提高非常重要。对于涉及非线性偏微分方程的模型的解释，即使在当前可用于数值分析的高计算能力的情况下，严格分析仍然保持着其不可或缺的作用。它不仅为其他艰巨的计算提供指导和简化，而且在许多情况下，定性分析是处理涉及非线性方程通解的难题的唯一方法。本项目解决的问题涉及解的几何特性（例如，当将其视为空间变量的函数时，它们的对称性）以及它们相对于时间的行为（周期性特性、平衡稳定性、所谓的有限时间内的爆炸） ）。开发解决这些问题的新数学技术是该项目的一个组成部分。