Singularity Formations in Nonlinear Elliptic and Parabolic Equations

奇点在非线性偏微分方程组中是普遍存在的。它们以尖锐的界面、漩涡、尖峰、有限或无限时间爆发等形式出现。我研究计划的长期目标是开发统一的方法来研究各种问题中的奇点形成。数学工具将包括粘合方法(有限或无限维)、非线性分析、几何和变分方法。建议的研究包括五个重叠的主题：I(De Giorgi猜想)：继续研究Allen-Cahn(AC)方程的De Giorgi猜想。在对单调解进行分类后，自然会对全局极小解、稳定的或有限的Morse指数解进行分类。然后利用它研究了Lane-Emden方程的De Giorgi型猜想。我们还对AC在构造测地线或黎曼流形上的极小曲面方面的应用感兴趣。另一个例子是超定问题，常平均曲率曲面在其中起着重要的作用。我们的最终目标是彻底解决Berestycki-Caffarelli-Nirenberg猜想。II(非局部方程)：研究非局部偏微分方程解的奇性，如分数阶Yamabe问题(存在性、紧性、奇异解)、分数极小曲面(稳定锥、极小图、叶)、半调和映射的第二类爆破、分数AC、分数粘合、行波到分数双稳AC的De Giorgi猜想III(第二类爆破)：发展分析抛物型方程第二类爆破的新方法，如调和映射流、Keller-Segel、具有临界或超临界非线性的非线性Fujita方程、欧拉方程。我们的目标是确定是否出现奇点，并描述井喷的性质，如井喷的速度、轮廓和稳定性特征。以前只分析了对称情况(径向)。我们的目标是开发通用技术来处理一般情况。IV(Toda系统)：用一般李代数对Toda系统的解进行分类和分析。以前的结果只给出了具有一个奇点的$A_n$Toda的分类。我们的重点将是对具有一般李群和具有多个奇点的Toda进行分类。然后，我们将利用它来研究流形上的Toda系统的非紧性，并计算它们的Leray-Schauder度。我们还构造了Chern-Simons-Higgs系统的鼓泡解和非拓扑解。V(反应扩散系统)：分析与反应扩散(RD)系统相关的新的局部化模式。新的观点包括：二维空间中格子解的稳定性、簇峰的存在、稳定性和连续极限、激活剂/抑制物中延迟的影响、RD几何对闭合流形的影响、非局部RD系统等。

Singularities are ubiquitous in nonlinear PDEs. They appear in the form of sharp interfaces, vortices, spikes, finite or infinite time blowup, etc. The long-term goal of my research program is to develop unified methods to study singularity formations in a diverse range of problems. The mathematical tools will include gluing methods (finite or infinite dimensional), nonlinear analysis, geometric and variational methods. The proposed research consists of five overlapping themes: I (De Giorgi Conjecture): Continue the study of De Giorgi conjectures for the Allen-Cahn (AC) equation. After the classification of monotone solutions, it is natural to classify global minimizers or stable or finite Morse index solutions. Then we use it to study De Giorgi type conjectures for Lane-Emden equations. We are also interested in the applications of AC to the constructions of geodesics or minimal surfaces on Riemannian manifolds. Another example is the over-determined problem for which constant mean-curvature surfaces play an important role. Our ultimate aim is to solve the Berestycki-Caffarelli-Nirenberg conjecture completely. II (Nonlocal Equations): Study singularities in nonlocal PDEs such as fractional Yamabe problems (existence, compactness, singular solutions), fractional minimal surfaces (stable cones, minimal graphs, foliations), Type II blowups for half-harmonic maps, the De Giorgi Conjecture for fractional AC, fractional gluing, travelling waves to fractional bistable AC.III (Type II Blow-up): Develop new techniques in analyzing Type II blow-ups in parabolic equations such as harmonic map flows, Keller-Segel, nonlinear Fujita equation with critical or supercritical nonlinearities, the Euler equation. Our goal is to determine whether or not singularities appear, and to describe the properties ofthe blowup, such as the rate, profile and stability character of blowups. Previously only symmetric cases (radial) have been analyzed. Our aim is to develop general techniques to deal with generic situations. IV (Toda Systems): Classify and analyze solutions of Toda systems with a general Lie algebra. Previous results only gave classification of $A_n$ Toda with one singularity. Our focus will be classification of Toda with a general Lie group and with more than one singularity. Then we shall use it to study non-compactness of Toda systems on manifolds and compute their Leray-Schauder degrees. We also want to construct bubbling solutions and non-topological solutions for Chern-Simons-Higgs systems. V (Reaction-Diffusion Systems): Analyze new localized patterns associated with reaction diffusion (RD) systems. New perspectives include: stability of lattice solutions in two dimensional space, existence, stability and continuum limits of clustered spikes, the effects of delays in theactivators/inhibitors, the effect of geometry for RD on closed manifolds, nonlocal RD systems, etc.