I am considering the following wave equation (for $\phi=\phi(x,t)$)
$$
\phi_{tt} - \Delta \phi =  \alpha |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}
$$
where $\nabla$ is just spatial gradient, i.e., $\nabla \phi= (\partial_{x_1} \phi, \partial_{x_2} \phi, \partial_{x_3} \phi)$ and $\alpha \in \mathbb{R}$. And assume that  initial data $\phi(x,0)$ and $\partial_t \phi(x,0)$  are regular enough.

This PDE in some sense is the complement case of the PDE in John's blow-up theorem, where we have
$$
\phi_{tt} - \Delta \phi =  (\partial_t\phi)^2.
$$


Now, I have the following questions: is this PDE well-posed? Or the solution will blow-up? Is there any paper on this PDE in 3+1 dimensions?