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, Birkha user, 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. 2000Academic Press

No abstract

No abstract

In this paper, we consider the initial value problems to systems of quasilinear wave equations with different speeds in two space dimensions. Applying JohnShatah observations to our problem, we introduce the null condition for the system with different speeds. Moreover, we prove a global existence theorem for a class satisfying the null condition.

No abstract