Qualitative Properties of Solutions of Nonlinear Elliptic and Parabolic Equations

本课题研究非线性抛物型和椭圆型偏微分方程，特别关注整个欧几里得空间中的问题。抛物线方程是进化方程吗？未知函数或解取决于一个或几个空间变量和另一个起时间作用的显著变量。这类方程广泛应用于应用科学的模型中。给定系统的初始状态，主要目标是描述其未来状态。从数学上讲，这转化为关于解中奇点可能发展的问题，以及在没有奇点的情况下，随着时间增加到无穷大，解的行为。有人问，解是否以某种方式接近于与时间无关的稳态，或者它是否可能表现出更复杂的行为。欧几里得空间上的椭圆方程代表了许多不同类型的演化偏微分方程的稳态（平衡）、孤波、行进锋或自相似解。因此，对椭圆方程的分析自然是理解这些演化方程动力学的关键基本步骤之一。这个项目吗?椭圆方程中的S问题涉及定性性质，如对称性、周期性和单个解的更复杂的振荡行为，以及解的全局结构，如它们的多重性和分岔（随着方程参数的变化而变化）。在本项目中进行的解的定性分析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非常重要。虽然该项目主要是理论性的，但其关于解的基本性质的结果可能对偏微分方程以外的研究领域感兴趣。例如，即使目前具有用于数值分析的高计算能力，如果没有定性分析的指导，涉及非线性偏微分方程的计算通常是艰巨的。此外，当研究应用科学中的特定偏微分方程模型时，关于给定类型方程解的可能行为的一般定性结果提供了有价值的信息。该项目包含研究生的组成项目和活动，该奖项提供研究生研究助理奖学金和夏季支持，并支持学生参加会议。这个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方程中，其中一个问题涉及解在除一个变量外的所有变量中都衰减为零。利用中心流形和KAM理论的技术，PI想要检验在非衰减变量中准周期解的存在性。另一类要考虑的椭圆方程是半线性热方程的自相似解的方程。PI将研究幂非线性中随指数变化的解的多重性及其从一个奇异解的分岔。对于实线上的抛物型方程，PI将继续研究其解在局域拓扑下的拟收敛性。对于整个空间上的多维半线性抛物方程，要解决的一个基本问题是，在足够高的空间维度上，在任何足够大的有界区域上，解是否可能在远离稳态的情况下表现出某种振荡行为。另外两个问题涉及非线性抛物方程全解的liouville型定理和分类定理（即在所有时间定义的解，正的和负的）。其中一个目标是证明指数的最优sobolev -次临界范围的正完整解的不存在性。另一方面，研究了超临界指数范围内的非平稳全解的存在性。Liouville定理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，其中一些如爆破类型的表征和空间爆破剖面的存在性等也属于本课题的预期成果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.