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

项目的一部分致力于研究各类具有对称性的抛物型偏微分方程组。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，而抛物型方程对称性问题的某些动机也源于这些定理(当把椭圆型方程的解看作相应抛物型方程的定态时)。其他非常有趣和具有挑战性的对称问题是抛物型方程所特有的。例如，当时间接近无穷大时，这是关于正解的渐近对称性的问题。首席研究员将继续他在这一领域的研究，目的是更好地了解正解如何逼近对称函数空间。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研究抛物型Liouville定理。这类定理表明，某些非常特殊的抛物型方程在一类可允许的函数中没有非平凡解。刘维尔定理对于抛物型方程的定性分析是非常有力的工具。与标度变元相结合，它们可用于先验估计的推导以及建立解的爆破和衰减率等。这个项目的目的是证明新的Liouville定理，并进一步探索Liouville定理在各类非线性抛物问题中的应用。刘维尔定理，以及关于解的对称性的结果，将在该项目的另一部分中发挥重要作用，这部分涉及阈值解。这类解表现为表现出两种不同行为的解之间的分离，例如在有限时间内衰减到零并爆炸，或者衰减到零并局部一致收敛到正稳态。例如，从燃烧理论和种群遗传学的角度研究了这类解与模型中的猝灭和传播现象有关的问题。到目前为止，已有的定理大多处理一维方程或具有变分结构的问题，从而排除了包含对流项或显式时间相关的重要方程。该项目将专注于这些非变分问题。用不那么专业的术语来说，这个项目可以被描述为对某种类型的非线性发展方程的解进行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。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、在有限时间内爆破)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。

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.