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。在最新发展的基础上，主要研究者将研究椭圆型偏微分方程解的对称性和非负解的节点结构。主要目的是对具有非平凡节点集的解可以存在的空间域进行分类，并确定对于空间齐次方程是否存在这样的解。在抛物型方程中，当时间增加到无穷大时，正解有“改善其对称性”的趋势，这是抛物线流动如何降低空间复杂性的一个显著例子。主要研究者将继续研究这一有趣的渐近对称现象，同时牢记渐近对称定理在抛物型方程收敛结果中的应用。还将使用其他方法来解决抛物型偏微分方程解收敛到平衡点的几个长期存在的问题。这个项目中的另一个主题是关于整个欧氏空间上椭圆型偏微分方程解的正解，这些正解在某些变量中衰减为零，但在其他变量中不衰减。我们将考察解相对于衰变变量的对称性以及它们相对于剩余变量的行为。主要研究者还将继续研究关于特定类型的非线性方程的非平凡解的不存在的Liouville类型的定理。基于Liouville定理的标度技术在抛物型偏微分方程组理论中有着广泛的应用，这将在本项目中得到进一步的探索。上述项目的成果将被应用于各种抛物型问题的阈值解的研究。这样的解作为两种不同行为的解之间的分离出现，例如在有限时间内衰减到零和爆破。它们的研究是纯粹出于理论上的原因，也是与应用科学中的猝灭和传播现象有关的。用非技术性的术语来说，这个项目可以被描述为对非线性偏微分方程解的定性分析或几何分析。这种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。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提高它们的模型相关性具有重要意义。对于涉及非线性偏微分方程的模型的解释，即使在目前可用于数值分析的高计算能力的情况下，严格分析仍保持其不可或缺的作用。它不仅为其他令人敬畏的计算提供指导和简化，而且在许多情况下，定性分析是处理与非线性方程一般解有关的困难问题的唯一方法。本项目涉及解的几何性质(例如，当它们被视为空间变量的函数时的对称性)以及它们在时间上的行为(周期性性质，稳定到平衡，所谓的有限时间爆破)的问题。为解决这类问题开发新的数学技术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.