In this paper we deal with systems of nonlinear wave equations in two space dimensions. When the system has common propagation speeds and cubic nonlinearity, the small data global existence result was obtained by Katayama , provided that the cubic part of Taylor's expansion for the nonlinearity satisfies the so-called null condition. The aim of this paper is to extend the result to the case where the system has multiple speeds of propagation. To realize this, we make use of a kind of Hardy's inequality given in Lemma 2.2 below, which creates the loss of decay but only with respect to $(1+||x|-c_i t|)$. Thus we are able to absorb such a loss by means of the decay estimates in Proposition 4.2 below.
"Global solvability for systems of nonlinear wave equations with multiple speeds in two space dimensions." Differential Integral Equations 17 (5-6) 593 - 622, 2004.