Global properties and large-time behavior of solutions nonlinear parabolic equations

非线性抛物型方程解的全局性质和大时间行为

• 批准号: 0900947
• 负责人: Peter Polacik
• 金额: $ 19.5万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900947&HistoricalAwards=false
• 关键词: Global; properties; large; time; behavior

A part of the project is devoted to the study of various classes of parabolic partial differential equations with symmetries. The basic question to be addressed is how positive solutions reflect the symmetry of the equation. For elliptic equations, there are classical theorems on symmetry of positive solutions and some motivations for symmetry problems in parabolic equations stem from these theorems (when viewing solutions of an elliptic equation as steady states of the corresponding parabolic equation). Other very interesting and challenging symmetry problems are specific to parabolic equations. Such are, for example, problems concerning the asymptotic symmetry of positive solutions as time approaches infinity. The principal investigator will continue his research in this area with the goal of obtaining a better understanding of the manner in which positive solutions approach the space of symmetric functions. Results in this vein would be instrumental in using the asymptotic symmetry of solutions to study their temporal behavior. The principal investigator will also continue his research concerning parabolic Liouville theorems. Such theorems state that certain very specific parabolic equations do not have nontrivial solutions in a class of admissible functions. Liouville theorems, when available, are very powerful tools for the qualitative analysis of parabolic equations. In combination with scaling arguments, they can be used, among other things, for the derivation of a priori estimates and for establishing blow-up and decay rates of solutions. The goal of this project is to prove new Liouville theorems and pursue further applications of Liouville theorems in various classes of nonlinear parabolic problems. Liouville theorems, as well as results on symmetry of solutions, will play important roles in another part of the project, which concerns threshold solutions. Such solutions appear as separatrices between solutions exhibiting two different kinds of behavior, such as the decay to zero and blow up in finite time, or decay to zero and locally uniform convergence to a positive steady state. Solutions of this type are studied, for example, in connection with quenching and propagation phenomena in models from combustion theory and population genetics. Up to now, existing theorems mostly treated one-dimensional equations or problems with a variational structure, thus excluding important equations that involve advection terms or explicit time dependence. The project will focus on these nonvariational problems. In less technical terms, the project can be characterized as qualitative or geometric analysis of solutions of a certain type of nonlinear evolution equations. Such equations are widely used in models in 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 improvement of their modeling relevance. 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, blow up in finite time). Development of new mathematical techniques for addressing such questions is an integral part of the project.

该项目的一部分致力于研究各种具有对称性的抛物型偏微分方程。要解决的基本问题是正解如何反映方程的对称性。对于椭圆型方程，有关于正解对称性的经典定理，抛物型方程对称性问题的一些动机源于这些定理（当将椭圆型方程的解视为相应抛物型方程的稳态时）。其他非常有趣和具有挑战性的对称问题是抛物线方程特有的。例如，当时间趋于无穷时，正解的渐近对称性问题。首席研究员将继续他在这一领域的研究，目标是更好地理解正解接近对称函数空间的方式。这方面的结果将有助于利用解的渐近对称性来研究它们的时间行为。首席研究员还将继续他关于抛物线刘维尔定理的研究。这些定理表明，在一类可容许函数中，某些非常特殊的抛物方程没有非平凡解。当可用时，刘维尔定理是对抛物方程进行定性分析的有力工具。与尺度论证相结合，它们可以用于推导先验估计，以及建立解的爆破率和衰减率。本课题的目标是证明新的Liouville定理，并进一步研究Liouville定理在各类非线性抛物问题中的应用。Liouville定理，以及关于解的对称性的结果，将在项目的另一部分中发挥重要作用，这一部分涉及阈值解。这样的解表现为解之间的分离，表现出两种不同的行为，如衰减到零并在有限时间内爆炸，或衰减到零并局部一致收敛到正稳态。例如，在燃烧理论和群体遗传学的模型中，研究了这种类型的解与淬火和繁殖现象的关系。到目前为止，现有的定理大多是处理一维方程或变分结构的问题，从而排除了涉及平流项或明确的时间依赖性的重要方程。该项目将重点关注这些非变分问题。在不那么技术性的术语中，该项目可以被描述为对某一类非线性演化方程的解进行定性或几何分析。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。理解解的定性性质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。目前的项目解决了有关解的几何性质的问题（例如当它们被视为空间变量的函数时的对称性），以及它们关于时间的行为（周期性性质，平衡稳定性，在有限时间内爆炸）。开发新的数学技术来解决这些问题是这个项目的一个组成部分。