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 and compactly supported.

This PDE in some sense is the complementary 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?

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

This PDE in some sense is the complementary 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?

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 complementary 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?

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 and compactly supported.

This PDE in some sense is the complementary 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?