Singularity Formations in Nonlinear Elliptic and Parabolic Equations

奇异性在非线性偏微分方程中是普遍存在的。它们以尖锐的界面、漩涡、尖峰、有限或无限时间爆炸等形式出现。我的研究计划的长期目标是发展统一的方法来研究奇点形成在不同范围的问题。数学工具将包括粘接方法（有限或无限维），非线性分析，几何和变分方法。提出的研究包括五个重叠的主题：I （De Giorgi猜想）：继续对Allen-Cahn （AC）方程的De Giorgi猜想的研究。在对单调解进行分类之后，自然要对全局极小解或稳定或有限摩尔斯指数解进行分类。然后用它研究了Lane-Emden方程的De Giorgi型猜想。我们也对AC在黎曼流形上测地线或极小曲面的构造中的应用感兴趣。另一个例子是过度确定问题，其中恒定平均曲率曲面起着重要作用。我们的最终目标是完全解决Berestycki-Caffarelli-Nirenberg猜想。II（非局部方程）：研究非局部偏微分方程中的奇异性，如分数阶Yamabe问题（存在性、紧性、奇异解）、分数阶极小曲面（稳定锥、极小图、叶状）、半调和映射的II型爆破、分数阶AC的De Giorgi猜想、分数阶AC的分数阶胶合、分数阶双稳AC的行波。开发分析抛物型方程中II型爆炸的新技术，如谐波映射流，Keller-Segel，具有临界或超临界非线性的非线性Fujita方程，欧拉方程。我们的目标是确定奇点是否出现，并描述爆炸的性质，如爆炸的速率、轮廓和稳定性特征。以前只分析了对称情况（径向）。我们的目标是发展处理一般情况的一般技术。IV （Toda系统）：用一般李代数对Toda系统的解进行分类和分析。以往的结果只给出了具有一个奇点的$A_n$ Toda的分类。我们将重点讨论具有一般李群和多个奇点的Toda的分类。然后我们将用它来研究流形上Toda系统的非紧性，并计算它们的Leray-Schauder度。我们还想构造chen - simons - higgs系统的冒泡解和非拓扑解。V（反应扩散系统）：分析与反应扩散（RD）系统相关的新的局部模式。新的观点包括：二维空间中晶格解的稳定性，簇尖的存在性、稳定性和连续极限，激活剂/抑制剂的延迟影响，几何形状对闭合流形的影响，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.