  For the 3D quasilinear wave equation
$-\big(1+(\partial_t\phi)^p\big)\partial_t^2\phi+\Delta\phi=0$ with the short
pulse initial data
$(\phi,\partial_t\phi)(1,x)=\big(\delta^{2-\varepsilon_0}\phi_0(\frac{r-1}{\delta},\omega),
\delta^{1-\varepsilon_0}\phi_1(\frac{r-1}{\delta},\omega)\big)$, where
$p\in\mathbb N$, $p\geq 2$, $0<\varepsilon_0<1$, $r=|x|$, $\omega=
\frac{x}{r}\in\mathbb S^2$, and $\delta>0$ is sufficiently small, under the
outgoing constraint condition
$(\partial_t+\partial_r)^k\phi(1,x)=O(\delta^{2-\varepsilon_0})$ for $k=1,2$,
we will establish the global existence of smooth large data solution $\phi$
when $p>p_c$ with $p_c=\frac{1}{1-\varepsilon_0}$ being the critical exponent.
In the forthcoming paper, when $1\leq p\leq p_c$, we show the formation of the
outgoing shock before the time $t=2$ under the suitable assumptions of
$(\phi_0,\phi_1)$.
