The lifespan of classical solutions to systems of nonlinear wave equations

Any results in this talk are based on a joint paper with R. Agemi & Y. Kurokawa [1]. The existence of the critical curve for p-q systems of nonlinear wave equations was already established by D. Del Santo & V. Georgiev & E. Mitidieri [3] except for the critical case. Our main purpose is to prove a blow-up theorem for which the nonlinearity (p, q) is just on the critical curve in three space dimensions. Moreover, the lower and upper bounds of the lifespan of solutions are precisely estimated including the sub-critical case.