Small Data Blowup for Systems of Semilinear Wave Equations with Different Propagation Speeds in Three Space Dimensions

We consider the Cauchy problem for a system of semilinear wave equations with small initial data and critical nonlinearity. As for a class of systems of quasilinear wave equations with critical nonlinearity, the small data global existence has been well developed for the case when the propagation speeds are distinct. In contrast with the quasilinear case, we show that the critical small data blowup occurs for the semilinear case, even if the propagation speeds are different from each other.