  We shall be concerned with the Cauchy problem for quasilinear systems in
three space dimensions of the form \label{i.1} \partial^2_tu^I-c^2_I\Delta u^I
= C^{IJK}_{abc}\partial_c u^J\partial_a\partial_b u^K + B^{IJK}_{ab}\partial_a
u^J\partial_b u^K, \quad I=1,..., D.
  Here we are using the convention of summing repeated indices, and $\partial
u$ denotes the space-time gradient, $\partial u=(\partial_0 u, \partial_1 u,
\partial_2 u, \partial_3u)$, with $\partial_0=\partial_t$, and
$\partial_j=\partial_{x_j}$, $j=1,2,3$. We shall be in the nonrelativistic case
where we assume that the wave speeds $c_k$ are all positive but not necessarily
equal.
  Using a new pointwise estimate of the M. Keel, H. Smith and the author we
shall prove global existence of small amplitude solutions for such equations
satisfying a null condition. This generalizes the earlier result of
Christodoulou and Klainerman where all the wave speeds are the same. Our
approach is related to that of Klainerman; however, since we are in the
non-relativistic case we cannot use the Lorentz boost vector fields or the
Morawetz vector fields. Instead we exploit both the 1/t decay of linear
solutions as well as the much easier to prove 1/|x| decay.

Analysis of PDEs