Global Solutions of Semilinear Parabolic and Elliptic Equations

半线性抛物型和椭圆方程的全局解

• 批准号: 9801271
• 负责人: Qi Zhang
• 金额: --
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801271&HistoricalAwards=false
• 关键词: Global; Solutions; Semilinear; Parabolic; Elliptic

Qi S. Zhang DMS-9801271 GLOBAL SOLUTIONS OF SEMILINEAR PARABOLIC AND ELLIPTIC EQUATIONS The proposed research is on the existence, nonexistence and singularity formations of global solutions to semilinear parabolic and elliptic equations and systems which arise naturally in many diverse fields such as geometry, chemistry, biology and physics. These equations include, among others, the equations of prescribed scalar curvatures, reaction diffusion equations and systems. The primary goal is to find conditions (optimal if possible) so that these equations shall have global positive solutions; solutions with sufficiently regular local behavior such as continuity; good asymptotic behavior when time becomes large; solutions that blow up in finite time. In the case solutions blow up, an effort will also be made to understand the pattern of singularities. The mathematical problems under investigations arise from model problems in several practical areas such as nonlinear heat transfer, biology, chemical reaction theory, physics. These natural phenomenons are described by certain semilinear or nonlinear partial differential equations. One of the central problems is to understand when or if solutions to these equations may be stable in the long run, and when or if solutions may blow up in finite time. These will give important information to the physical models as whether a chemical reaction may blow up or stabilize in long time, or whether shocks may be formed in wave propagations, and whether or not two competing species may co-exist.

Qi S.Zhang DMS-9801271 半线性抛物型和椭圆方程的全局解所提出的研究是关于半线性抛物型和椭圆方程和系统的全局解的存在、不存在和奇异性形成，这些解在几何、化学等许多不同领域中自然出现、生物学和物理学。 这些方程包括规定标量曲率方程、反应扩散方程和系统等。 主要目标是找到条件（如果可能的话是最优的）使得这些方程具有全局正解；具有足够规律的局部行为（例如连续性）的解决方案；当时间变大时，具有良好的渐近行为；在有限时间内爆炸的解决方案。 在解决方案爆炸的情况下，还将努力理解奇点的模式。 所研究的数学问题源自非线性传热、生物学、化学反应理论、物理学等多个实际领域的模型问题。 这些自然现象是通过某些半线性或非线性偏微分方程来描述的。 核心问题之一是了解这些方程的解何时或是否可以长期稳定，以及解何时或是否可以在有限时间内爆炸。 这些将为物理模型提供重要信息，例如化学反应是否会爆发或长时间稳定，或者波传播中是否会形成激波，以及两个竞争物种是否可以共存。