Semilinear hyperbolic systems and equations with singular initial data

We study the weak limits (t, .)of solutions to semilinear strictly hyperbolic systems and wave equations with initial datau(0, .) approximating a distribution, 0 < 1. We propose an optimal link between the singularity of and the growth of the nonlinear term in order that exists. In this way we extend some of the results in [3], [10], [13].