Global Solutions of Semilinear Parabolic and Elliptic Equations

Qi S. Zhang DMS-9801271 半线性抛物型和椭圆型方程的整体解 拟议的研究是在几何，化学，生物学和物理学等许多不同领域自然产生的半线性抛物和椭圆方程和系统的整体解的存在性，不存在性和奇异性的形成。 这些方程包括，除其他外，规定的标量曲率方程，反应扩散方程和系统。 主要目标是找到条件（如果可能的话是最优的），使得这些方程应该有全局正解;具有足够规则的局部行为（如连续性）的解;时间变大时的良好渐近行为;在有限时间内爆破的解。 在解爆炸的情况下，我们还将努力理解奇点的模式。 在调查中的数学问题产生的模型问题，在几个实际领域，如非线性传热，生物学，化学反应理论，物理学。 这些自然现象可以用半线性或非线性偏微分方程来描述。 其中一个核心问题是了解这些方程的解何时或是否在长期内是稳定的，以及解何时或是否在有限时间内爆炸。 这些将为物理模型提供重要信息，例如化学反应是否会在长时间内爆炸或稳定，或者在波传播中是否会形成冲击，以及两个竞争物种是否会共存。

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.