Development of singularities for nonlinear hyperbolic systems with periodic data

We prove two blow-up results for classical (C ¹) solutions of the Cauchy problem for nonlinear hyperbolic systems in one space variable. The initial conditions are periodic and the systems are supposed to be in diagonal form. The first result concernsN×N genuinely nonlinear systems, and the second one is devoted to 2×2 systems with weak nonlinearities. In both cases we give an estimate of the lifespan of the classical solutions.