Qualitative Properties of Solutions of Nonlinear Elliptic and Parabolic Equations

这个项目是关于非线性抛物型和椭圆型偏微分方程组，特别关注在整个欧氏空间上提出的问题。抛物型方程是演化方程--未知函数或解取决于一个或几个空间变量，以及一个或多个起时间作用的显著变量。这样的方程在应用科学的模型中被广泛使用。给定系统的初始状态，主要目标是描述其未来状态。在数学上，这转化为关于解中奇点的可能发展的问题，以及在没有这种奇点的情况下，随着时间增加到无穷大，解的行为的问题。有人问，解是否以某种方式接近与时间无关的稳态，或者它是否可能表现出更复杂的行为。欧氏空间上的椭圆型方程表示许多不同类型的发展偏微分方程解的定态(平衡)、孤立波、行波或自相似解。因此，分析椭圆型方程自然是理解这些演化方程动力学的关键基本步骤之一。项目？？椭圆型方程中的S问题涉及定性性质，如对称性、周期性，以及更复杂的解的振荡行为，以及解的整体结构，如其多重性和分支(随着方程中参数的变化而变化)。对本项目中将要进行的解决方案进行定性分析，对于发展偏微分方程组的数学理论以及改进其模型相关性具有重要意义。虽然该项目主要是理论上的，但它关于解的基本性质的结果可能会在偏微分方程组以外的研究领域引起兴趣。例如，即使有目前可用于数值分析的高计算能力，涉及非线性偏微分方程组的计算通常在没有定性分析的指导下也是令人敬畏的。此外，当要研究来自应用科学的特定偏微分方程组模型时，关于给定类型的方程的解的可能行为的一般定性结果提供了有价值的信息。该项目包括研究生项目和活动的组成部分，该奖项为研究生提供研究助学金和暑期支持，并支持学生参加会议。该项目的研究将沿着几个主要主题发展。在整个空间上的某些椭圆型方程中，其中一个问题涉及到除一个变量外所有变量都衰减到零的解。利用中心流形和KAM理论的技巧，PI想要检查非衰变变量中准周期解的存在性。另一类需要考虑的椭圆型方程作为半线性热方程的自相似解的方程出现。PI将研究随着功率非线性指数的变化，奇异解的多解性及其分叉。对于实线上的抛物型方程，PI将继续他关于解关于局部拓扑的拟收敛性质的研究。对于整个空间上的多维半线性抛物型方程，需要解决的一个基本问题是，在足够高的空间维度上，解是否会在任何足够大的有界区域上表现出某种振荡行为，同时远离定态。另外两个问题涉及到非线性抛物型方程的整体解(即所有时刻定义的解，正解和负解)的Liouville类型和分类定理。其中一个目标是证明指数的最优Sobolev-亚临界范围的正整体解的不存在性。在另一个例子中，研究了超临界指数范围内非平稳整体解的存在性。刘维尔和分类定理在抛物方程解的爆破理论中有许多有趣的应用，其中一些，例如表征爆破的类型和空间爆破轮廓的存在，也属于该项目的预期结果。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with nonlinear parabolic and elliptic partial differential equations (PDEs), with special focus on problems posed on an entire Euclidean space. Parabolic equations are evolution equations?the unknown function, or the solution, depends on one or several spatial variables and one more distinguished variable playing the role of time. Such equations are widely used in models in applied sciences. Given an initial state of the system, the main goal is to describe its future states. Mathematically, this translates to questions about a possible development of singularities in the solutions, and, in the absence of such singularities, about the behavior of the solutions as time increases to infinity. One asks if the solution approaches in some way a time-independent steady state or if it may exhibit a more complicated behavior. Elliptic equations on Euclidean spaces represent steady states (equilibria), solitary waves, traveling fronts, or self-similar solutions of many different types of evolution PDEs. Naturally, therefore, analysis of elliptic equations is one of the key basic steps toward understanding of the dynamics of these evolution equations. The project?s problems in elliptic equations concern qualitative properties, such as symmetry, periodicity, and more complex oscillatory behavior of individual solutions, as well as the global structure of the solutions, such as their multiplicity and bifurcations (changes as parameters in the equation vary). Qualitative analysis of solutions to be carried out in this project is important for the internal development of the mathematical theory of PDEs as well as for improvement of their modeling relevance. Although the project is mainly theorical, its results concerning fundamental properties of solutions could be of interest in research fields beyond PDEs. For example, even with high computing power currently available for numerical analysis, computations involving nonlinear PDEs are often formidable without a guideline from qualitative analysis. Also, when a specific PDE model from applied science is to be investigated, general qualitative results on possible behavior of solutions of equations of the given type provide a valuable information. This project contains component projects and activities for graduate students, and the award provides graduate student research assistantship and summer support, and support for student participation at conferences.The research in this project will develop along several main topics. In certain elliptic equations on the entire space, one of the problems concerns solutions which decay to zero in all but one variable. Employing techniques from the center manifold and KAM theories, the PI wants to examine the existence of solutions which are quasiperiodic in the non-decay variable. Another class of elliptic equations to be considered arises as an equation for self-similar solutions of the semilinear heat equation. The PI will study the multiplicity of the solutions and their bifurcations from a singular solution as the exponent in the power nonlinearity varies. For parabolic equations on the real line, the PI will continue his research on quasiconvergence properties of solutions with respect to a localized topology. For multidimensional semilinear parabolic equations on the entire space, one of the basic questions to be addressed is whether in high enough spatial dimensions, the solutions may exhibit some sort of oscillatory behavior while staying away from steady states on any sufficiently large bounded region. Two other problems deal with Liouville-type and classification theorems for entire solutions (that is, solutions defined for all times, positive and negative) of nonlinear parabolic equations. In one of them, the goal is to prove the nonexistence of positive entire solutions for an optimal Sobolev-subcritical range of exponents. In another one, the existence of nonstationary entire solutions is to be investigated for a range of supercritical exponents. Liouville and classification theorems have many interesting applications in the theory of blowup of solutions of parabolic equations and some of them, such as the characterization the type of blowup and existence of spatial blowup profiles, also belong to the expected outcome of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.