9401558 Shatah This award supports mathematical research on partial differential equations which arise in studies of classical fields. The work involves studies of hyperbolic Yang-Mils equations and wave maps. It is planned to address fundamental questions on existence, uniqueness and formation of singularities for these models. Also considered will be models defined on the n-sphere where efforts will be made to show that, at least in the one-dimensional case, unstable stationary solutions possess homoclinic orbits. Analysis of the asymptotic behavior of solutions to perturbed wave equations and nonlinear lattices will also be carried out. Decay estimates for solutions of the wave equation on the Schwarzschild metric where there is a trapped sphere and decay rates for the discrete wave equation will be developed. This results will be used to study the dynamics of nonlinear multi-dimensional lattices, specifically the phenomenon of resonance. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***
