Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponent under the null condition

This paper deals with the Cauchy problems of nonlinear hyperbolic systems i n two space dimensions with small data. We assume that the propagation spee ds differ from each other and that nonlinearities are cubic. Then it will b e shown that if the nonlinearities satisfy the null condition, there exists a global smooth solution. To prove this kind of claim, one usually makes u se of the generalized differential operators Omega(ij), S, and L-i, which w ill be introduced in section 1. But it is difficult to adopt the operators L-i = x(i)partial derivative(t) + t partial derivative x(i) to our problem, because they do not commute with the d'Alembertian whose propagation speed is not equal to one. We succeed in taking L-i away from the proof of our t heorem. One can apply our method to a scalar equation; hence L-i are needle ss in this kind of argument.