The work undertaken in this project concerns partial differential equations of hyperbolic type. These equations are frequently used to model wave phenomena occurring the physical universe. Examples can be found in the theories of elasticity, elementary particle physics and general relativity. The general paradigm for studying a class of hyperbolic equations is to show first the existence of what are known as weak solutions, to establish properties of these solutions and then examine the large time behavior of the solutions. In the last context, one looks for stationary or static solutions through a conversion of the (now time-independent) equation to elliptic form. This research will focus on nonlinear waves described by equations whose primary nonlinear term (as distinct from the fundamental wave equation) is a power of the modulus of the solution. At issue is whether solutions can be shown to exist for all time or will singularities develop? It turns out that a nonlinearity equal to the fifth power of the solution is the critical exponent for existing theory. For all values less than five, regularity obtains. Recent work provides proof of the regularity even at the critical value. Current work will now be confined to equations involving higher powers of the nonlinearity. Past methods will definitely not work here. Other investigations will be carried out on the wave equation in curved space-time to determine whether the energy of the solution decays locally in space and on Hamiltonian systems that are invariant under the action of a Lie group. Analysis of these systems will be directed toward a better understanding of the stability of periodic solutions.
