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.
