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

该项目的一部分是致力于研究各类抛物型偏微分方程的对称性。 要解决的基本问题是积极的解决方案如何反映 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，抛物型方程中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性问题是抛物方程特有的。例如，当时间接近无穷大时，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，以获得更好的理解的方式，其中正解的方法空间的对称函数的目标。 在这方面的结果将有助于使用的渐近对称性的解决方案，以研究其时间行为。 首席研究员还将继续他的研究有关抛物刘维尔定理。 这样的定理指出，某些非常特殊的抛物型方程在一类容许函数中没有非平凡解。 刘维尔定理是抛物型方程定性分析的有力工具。 与尺度参数相结合，它们可以用于推导先验估计和建立解的爆破和衰减速率。这个项目的目标 就是证明新的刘维尔 定理和 进一步研究刘维尔定理在各类非线性抛物问题中的应用。刘维定理，以及解决方案的对称性的结果，将发挥重要作用，在该项目的另一部分，这涉及阈值的解决方案。这样的解决方案之间的解表现出两种不同的行为，如衰减到零和爆破在有限时间内，或衰减到零和局部一致收敛到一个积极的稳定状态的分离线出现。这种类型的解决方案的研究，例如，在 与燃烧理论和群体遗传学模型中的猝灭和传播现象有关。到目前为止，已有的定理大多是处理一维方程或具有变分结构的问题，从而排除了涉及 平流 方面 或显式 时间依赖性这个项目将集中在这些非变分问题。 用不太专业的术语来说，该项目可以被描述为对某种类型的非线性演化方程的解进行定性或几何分析。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。 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，在有限时间爆破）。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。

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.