Non-Existence of Global Solutions of partial of F(u sub t) with Respect to t in Two and Three Space Dimensions

摘要

Solutions of nonlinear hyperbolic partial differential equations often develop singularities spontaneously. Physically this phenomenon corresponds to the formation of shocks in nonlinear waves. One is confronted with the questions: What are the factors contributing to this blow-up of solutions. How long does it take for blow-up to develop (i.e. what is the life span T of the solution). What goes on precisely during blow-up. There is no general answer covering the great variety of situations encountered. A critical role certainly is played by the size of the initial disturbance that gives rise to the wave solution, and by the number of dimensions of the space in which the wave propagates. One finds that larger disturbances are more likely to result in shocks, and that, on the other hand, with increased dimension there are more possibilities for the wave to spread out and to decay, thus counteracting the formation of shocks. This document is concerned with a special type of second order nonlinear wave equation, whose behavior can be expected to be typical for a large class of equations occurring in applications, e.g. in the propagation of waves of finite amplitude in elastic materials.