The Cauchy problem for quasi–linear hyperbolic evolution problems with a singularity in the time

Abstract

Consideration is given to abstract quasi–linear first order hyperbolic evolution problems with a forcing term having an initial 1/ t singularity in the time t . The results are obtained in the context of a reflexive Banach space. Existence and uniqueness of solutions are established in spaces of sufficiently t –differentiable functions, and stability with respect to perturbations of the initial data is proved. The special case of symmetric hyperbolic systems of first order PDEs is discussed. The work completes a previous study by Newman concerning Penrose9s Weyl Curvature Hypothesis in general relativity.