Asymptotics, Existence, Symmetry and Uniqueness in NonlinearPartial Differental Equations

摘要本文研究了一类非线性椭圆型和抛物型方程及系统解的存在性、定性性质（如渐近性、对称性等）和唯一性问题。更具体地说，将致力于研究Lane - Emden方程和一般非线性椭圆方程和系统的正解的渐近行为和对称性，Lane - Emden系统、标量场系统和半线性哈密顿系统的正解的存在性和不存在性，Ginzburg-Landau方程解的渐近性，研究一类热敏电阻问题解的存在性和正则性，以及半线性椭圆型方程正解的唯一性，重点讨论了区域几何形状的影响。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，并且，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各种非线性现象的研究，包括模式形成和种群进化（数学生物学），天体物理学和超导（物理学）中的恒星结构，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基本和必要的，这是数学分析的核心。这可能包括回答关于解的唯一性、光滑性、形状和渐近行为的问题。本提案中提出的调查就是为我们的模型问题提供这样的研究。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

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.