Global existence and blow-up of the classical solutions to systems of semilinear wave equations in three space dimensions

Abstract

We consider the Cauchy problem for a system of semilinear wave equations with small initial data whose propagation speeds may be different. As for a system of quasilinear wave equations, the discrepancy of the speeds makes the maximal existence time of solutions be longer, when we treat the critical nonlinearity. In contrast with the quasilinear case, we show that for the semilinear case, such phenomena does not occur, by establishing estimates of the lifespan from upper and lover.