Global Solutions of Semilinear Parabolic and Elliptic Equations

半线性抛物型和椭圆型方程组整体解的存在性、不存在性和奇性形式的研究自然地出现在几何、化学、生物和物理等许多不同的领域。这些方程包括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(如果可能是最优的)，使这些方程有全局正解；具有充分规则的局部行为的解，如连续性；当时间变大时良好的渐近行为；在有限时间内爆炸的解。在解决方案爆炸的情况下，还将努力理解奇点的模式。所研究的数学问题来源于几个实际领域的模型问题，如非线性传热学、生物学、化学反应理论、物理学。这些自然现象可用一定的半线性或非线性偏微分方程组来描述。核心问题之一是了解这些方程的解何时或是否长期稳定，以及解何时或是否可能在有限时间内爆炸。这些将为物理模型提供重要的信息，如化学反应是否可能在很长一段时间内爆炸或稳定，或是否可能在波的传播中形成冲击，以及两个相互竞争的物种是否可能共存。

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.