Critical Curve for p-q Systems of Nonlinear Wave Equations in Three Space Dimensions

The existence of the critical curve for p-q systems for nonlinear wave equations was already established by D. Del Santo, V. Georgiev, and E. Mitidieri [1997, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in "Geometric Optics and Related Topics" (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117-139, Birkhäuser, Basel] 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.