Global existence for systems of nonlinear wave equations in two space dimensions .2.

We consider the Cauchy problem for the system of nonlinear wave equations (*) (partial derivative(t)(2)-Delta)u(i) = F-i(u,u',u '') in (0,infinity)XR(2), i = 1,... N with initial data u(i)(0,x) = epsilon phi(i)(x), (partial derivative(t)u(i))(0,x) = epsilon psi(i)(x), where F-i(i = 1, ..., N) are smooth functions of degree 3 near the origin (u,u',u '') = 0, phi(i), psi(i) is an element of C-0(infinity)(R(2)) and epsilon is a small positive parameter. We assume that F-i(i = 1, ..., N) are independent of u(j)(infinity) for any j not equal i. In the previous paper, the author showed the global existence of the small solution to the Cauchy problem (*) assuming that the cubic parts of the nonlinear terms satisfy Klainerman's null condition and that the nonlinear terms are independent of u(j)u(k)u(l)u(m) for any j,k,l,m = 1, ..., N. In this paper, we show the global existence without imposing further assumptions than the null condition;on the cubic parts of the nonlinear terms.