8902136 Beals This project will emphasize the analysis of solutions of nonlinear strictly hyperbolic partial differential equations. Specifically, work will be done in constructing examples in which precislely described singularities are shown to exist, and to establish regularity results which show that these are essentially the worst possible singularities. Natural conditions on the smmoothness of the solution as it evolves in time are known to curtail the formation of singularities, at least locally in time. Among the related problems to be considered in this work are questions concerning interactions of singularities occurring in cases where caustic surfaces appear. This contrasts with that of simple interactions of singularities conormal across smooth characteristic surfaces which are well understood. In addition to this, the question of triple interactions will be addressed. It is known that piecewise smoothness of solutions in the past is not preserved after a triple interaction. A procedure for constructing solutions with given data of conormal or classical conormal type will be one of the objectives of this research. In four dimensions, simple initial conditions are expected to give rise to complex sets of nonlinear singularities. The construction of an example of this type of behavior will be treated. The wave equation on the exterior of a convex obstacle will also be considered. It represent the simplest problem involving interaction near a grazing surface of singularities; it may give rise to single nonlinear singularity. To accomplish some of these constructions, precise regularity theorems will be required to show that explicit approximate solutions differ from actual ones by remainders of strictly greater smoothness and to extend known regularity results to lower order smoothness.
