Singularity Formations in Nonlinear Elliptic and Parabolic Equations

非线性椭圆方程和抛物线方程中的奇异性形成

• 批准号: RGPIN-2018-03773
• 负责人: Wei, Juncheng
• 金额: $ 5.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758598
• 关键词: Singularity; Formations; Nonlinear; Elliptic; Parabolic

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.

奇异性在非线性PDE中无处不在。它们以尖锐的接口，涡流，尖峰，有限或无限的时间爆炸等形式出现。我的研究计划的长期目标是开发统一的方法来研究各种问题范围内的奇异性形成。 数学工具将包括胶合方法（有限或无限尺寸），非线性分析，几何和变分方法。 拟议的研究由五个重叠的主题组成：I（De Giorgi猜想）：继续研究Allen-Cahn（AC）方程的De Giorgi猜想。分类单调溶液后，自然将全球最小化器或稳定或有限的摩尔斯索引解决方案进行分类是很自然的。 然后，我们使用它来研究de giorgi类型的猜想。我们还对AC在Riemannian歧管上的地球学或最小表面的构造中的应用感兴趣。 另一个例子是过度确定的问题，恒定均曲面表面起着重要作用。我们的最终目的是完全解决Berestycki-Caffarelli-Nirenberg的猜想。 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（II型爆炸）：在分析抛物线​​方程中的II型爆破中开发新技术，例如谐波图流，凯勒 - 塞格，非线性富士夫，具有关键或超临界非线性，Euler方程。我们的目标是确定是否出现奇异性，并描述爆炸的特性，例如爆炸的速率，轮廓和稳定性。以前仅分析了对称病例（径向）。我们的目的是开发一般技术来处理通用情况。 IV（TODA系统）：将TODA系统的解决方案与一般谎言代数进行分类和分析。先前的结果仅给出了一个奇异性的$ a_n $ toda的分类。我们的重点是将TODA与一个普通的谎言组和多个奇异性分类。 然后，我们将使用它来研究TODA系统在流形上的非紧密度，并计算其Leray-Schauder学位。我们还希望为Chern-Simons-Higgs系统构建冒泡的解决方案和非亲人解决方案。 V（反应扩散系统）：分析与反应扩散（RD）系统相关的新局部模式。新的观点包括：晶格解决方案在二维空间中的稳定性，存在，稳定性和簇状尖峰的连续性极限，造物剂/抑制剂中延迟的影响，RD几何形状对封闭歧管，非局部RD系统等的影响