Suppose $x\in\mathbb{R}^n$, $V(x)$ and $\phi(x)$ are maps from $\mathbb{R}^n\rightarrow \mathbb{R}$, in which $\phi(x)$ is an auxiliary function.
Could anyone help me to classify this system of partial differential equations (It is called Zubov's equation)?
$$\nabla V(x)\cdot f(x)=\phi(x)(1-V(x))$$
My thought is that
(1) It is a first order pde system Is that correct?
(2) It is linear pde system. Is that correct?
(3) Is it an Elliptic or Hyperbolic or and Parabolic PDE?
(4) Is it a Haminton-Jabobi equation?
(5) Do you see other aspects of the system?
(6) how to solve this system?
Thanks in advance!