Asymptotics, Existence, Symmetry and Uniqueness in NonlinearPartial Differental Equations

非线性偏微分方程的渐近性、存在性、对称性和唯一性

基本信息

  • 批准号:
    9622937
  • 负责人:
  • 金额:
    $ 6.27万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    1996
  • 资助国家:
    美国
  • 起止时间:
    1996-08-01 至 2000-07-31
  • 项目状态:
    已结题

项目摘要

Abstract Zou This proposal is concerned with problems of existence, qualitative properties (e.g., asymptotics, symmetry, etc.) and uniqueness of solutions for some model nonlinear elliptic and parabolic equations as well as systems. More specifically, efforts will be devoted to investigations of the asymptotic behavior and symmetry of positive solutions for the Lane - Emden equation and general nonlinear elliptic equations and systems, the existence and non-existence for positive solutions of the Lane - Emden system, the scalar field system and a semilinear Hamiltonian system, the asymptotics of solutions of Ginzburg-Landau equations, the existence and regularity of solutions to a thermistor problem and the uniqueness of positive solutions of semilinear elliptic equations with emphasis on the impact of the geometry of the domain. Systematic methods and approaches for general problems will be developed by studying these model equations. Tools for this project include a variational approach, the moving plane method, the topological fixed point theory and degree theory, and, asymptotic and a priori estimates are essential. Partial differential equations form a basis for mathematical modeling of the physical world. Our model problems in this proposal arise from the study of various nonlinear phenomena, including pattern formation and population evolution (mathematical biology), the stellar structure in astrophysics and superconductivity (physics), the thermistor problem (industrial mathematics) and the Yamaba problem (differential geometry). It is, in studying classical partial differential equations, fundamental and essential to provide qualitative and quantitative information about the solutions, which lies in the core of mathematical analysis. This may include answers to questions about uniqueness, smoothness, shape and asymptotic behavior of solutions. The proposed investigations in this proposal are to provide such studies for our model problems. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.
摘要邹 这个建议涉及的问题的存在,定性性质(例如,渐近性、对称性等)一些模型非线性椭圆型和抛物型方程及方程组解的唯一性。 更具体地说,我们将致力于研究Lane -埃姆登方程和一般非线性椭圆型方程和方程组正解的渐近性态和对称性, Lane -埃姆登系统、纯量场系统和半线性Hamilton系统正解的不存在性、Ginzburg-Landau方程解的渐进性、热敏电阻问题解的存在性和正则性以及半线性椭圆方程正解的唯一性,重点讨论了区域几何的影响。 通过研究这些模型方程,将发展出解决一般问题的系统方法和途径。 这个项目的工具包括变分方法,移动平面方法,拓扑不动点理论和度理论,渐近和先验估计是必不可少的。 偏微分方程是物理世界数学建模的基础。我们的模型问题来自于对各种非线性现象的研究,包括模式形成和种群演化(数学生物学),天体物理学和超导学中的恒星结构(物理学),热敏电阻问题 (工业数学)和Yamaba问题(微分几何)。 在研究经典偏微分方程时,提供解的定性和定量信息是最基本和最重要的,这是数学分析的核心。 这可能包括解答有关解的唯一性、光滑性、形状和渐近行为的问题。 本提案中提出的调查是为我们的模型问题提供这样的研究。 此外,分析经常发展出解的近似方法和对这些近似的准确性的估计。

项目成果

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Henghui Zou其他文献

A priori estimates and existence for quasi-linear elliptic equations
A Priori Estimates and Existence for Strongly Coupled Semilinear Cooperative Elliptic Systems
On the well-posedness of a mathematical model for lithion-ion battery system
锂离子电池系统数学模型的适定性研究
EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS
On global existence for the Gierer-Meinhardt system

Henghui Zou的其他文献

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{{ truncateString('Henghui Zou', 18)}}的其他基金

Mathematical Sciences: Studies on Nonlinear Partial Differential Equations
数学科学:非线性偏微分方程研究
  • 批准号:
    9418779
  • 财政年份:
    1994
  • 资助金额:
    $ 6.27万
  • 项目类别:
    Standard Grant

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