The first step in the proof is to reduce the problem to the case when the initial data is small, using scaling and the finite speed of propagation. This is a routine argument for which we refer the reader to [@MR2178963]. Once we know that the initial data is small, we can fix the time interval and set it to $[-1,1]$. This will be the case throughout the rest of the paper.

To solve the problem for small data we use a fixed point argument. Let $S(u_0,u_1)$ and $\Box_{g}^{-1}$ be respectively the homogeneous and inhomogeneous solution operators $$\begin{aligned}
&\Box_{g}S(u_0,u_1)\,=\,0,\quad S(u_0,u_1)(0)\,=\,u_0,\quad
\partial_t S(u_0,u_1)(0)\,=\,u_1
\label{hom}\\
&\Box_{g}(\Box_{g}^{-1}H)\,=\,H,\qquad
(\Box_{g}^{-1}H)(0)\,=\,0,\qquad
\partial_t(\Box_{g}^{-1}H)(0)\,=\,0
\label{inhom}\end{aligned}$$ Then a solution $u$ for (\[eq\]) in $[-1,1]$ is also a fixed point for the functional $$F(u)\,= S(u_0,u_1) + \Box_{g}^{-1}( \Gamma(u)(\nabla
u)^2)
\label{funct}$$

In order to apply a fixed point argument for $F$ we need to find two Banach spaces $X$ and $Y$ for which the following mapping properties hold: $$\begin{aligned}
&\,\| S(u_0,u_1) \|_X \lesssim \|(u_0,u_1)\|_{H^s\times H^{s-1}}
\label{h}\\
&\,\|\Box_g^{-1}H\|_X \lesssim \|H\|_Y
\label{i}\\
&\,\|u \cdot w\|_X \lesssim \|u\|_X \|w\|_X
\label{xx}\\
&\|\Gamma(u)\|_{X} \lesssim C(\|u\|_{L^\infty}) (1 + \|u\|_{X}^5)
\label{moser} \\
&\,\|u \cdot w\|_Y \lesssim \|u\|_X \|w\|_Y
\label{xy} \\
&\,\|\nabla v\cdot \nabla w\|_Y\lesssim  \|v\|_X\cdot \|w\|_X
\label{n}\end{aligned}$$ where 