This project continues and expands upon earlier work focusing on the formulation of singularities in solutions of nonlinear hyperbolic equations. In particular recent new developments related to systems will be continued. The immediate aims are to obtain asymptotic estimates for the time when the first singularity occurs. This point is frequently referred to as the moment of blow-up. Initial conditions involve data of compact support of fixed shape and size. The immediate aim of this work is to obtain asymptotic estimates for the time at which blow-up occurs based on the initial size. Earlier work on the nonlinear wave equation and systems of elasticity equations only produced lower bounds for this critical time. Projected work will be concerned with the derivation of supplementary upper limits which would, in turn, imply blow-up in finite time. In addition to these estimates work will be done in analyzing the behavior of solutions near blow-up time. While these questions are relatively simple in one space dimension, the situation for general three-dimensional solutions is much more complicated. One will have to analyze the extent to which higher angular derivatives may eliminate contributions from lower radical ones. Efforts will also be made to understand long-term behavior of quasilinear systems which reduce to homogeneous linear systems with constant coefficients for infinitesimal solutions. An example of this type of problem arises in the study of finite amplitude waves in an unistropic hyperelastic material.
