| This paper is concerned with the Cauchy problem for a system of quadratically nonlinear wave equations in three space dimensions, with different propagation speeds and sufficiently small initial values. By directly estimating the fundamental solution to the system of equations, it was shown that under some nonresonance condition, or the so-called null condition, imposed on the quadratic portion of the nonlinearity, the system of wave equations posses existence of global strong solution. In order to avoid direct estimation of the fundamental solution to the system of wave equations, we will give a simplified proof using only the classical invariance of the equations under rotations, translations and changes of scales. The existence proof is based on the idea of energy estimates with respect to the generators of the invariants. The proof also relies on weighted Linfinity--L 2 Sobolev-type estimates and weighted L2-- L2 inequalities. |
