Qualitative studies of solutions of nonlinear elliptic and parabolic equations

该项目致力于偏微分方程（PDE）的定性研究。基于最近的发展，主要研究者将研究椭圆型偏微分方程非负解的对称性和节点结构。主要目标是对空间域进行分类，在空间域上可以存在具有非平凡节点集的解，并确定空间齐次方程是否存在这样的解。在抛物型方程中，正解随着时间的增加而“改善其对称性”的趋势是抛物流如何降低空间复杂性的一个显着例子。主要研究人员将继续他的研究这个有趣的渐近对称现象，同时铭记渐近对称定理的应用在收敛结果抛物方程。其他方法也将被用来解决一些长期存在的问题，抛物型偏微分方程的解决方案的收敛到平衡。本计画的另一个主题是关于整个欧几里德空间上椭圆型偏微分方程的正解，在某些变数下衰减为零，但在其他变数下不衰减。我们将研究解相对于衰变变量的对称性及其相对于其余变量的行为。首席研究员也将继续他的研究有关刘维尔型定理的非平凡解决方案的特定类别的非线性方程的不存在性。基于刘维尔定理的尺度技术在抛物型偏微分方程理论中有着广泛的应用，这将在本项目中进一步探讨。上述项目的结果将应用于各种抛物问题的阈值解的研究。这样的解决方案之间出现的分歧，表现出两种不同的行为，如衰减到零和爆破在有限的时间。它们的研究纯粹是出于理论上的原因，也与应用科学中的猝灭和传播现象有关。用不那么专业的术语来说，这个项目可以被描述为非线性偏微分方程解的定性或几何分析。这些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以及改进其建模相关性是重要的。对于涉及非线性偏微分方程的模型的解释，严格的分析保持其不可或缺的作用，即使在存在的高计算能力，目前可用于数值分析。它不仅提供了指导方针和简化，否则艰巨的计算，但在许多情况下，定性分析是唯一的方法来处理困难的问题，一般解决非线性方程。本项目解决的问题涉及的几何性质的解决方案（如他们的对称性，当被视为空间变量的函数），以及他们的行为相对于时间（周期性的属性，稳定的平衡，所谓的爆破在有限时间）。开发新的数学技术来解决这些问题是该项目的一个组成部分。

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.