So I have a system of PDE's essentially consisting of equations of the form $$\sum_{i=1}^{n}f_{i}(\mathbf{x}(t), t)\frac{\partial u(\mathbf{x}(t),t)}{\partial x_i}=c$$ where $c$ is any nonzero constant and $\mathbf{x}(t)$ is a state vector of Itô processes.
I was wondering if anything can be said about what type of PDE this is and what determines the existence of a solution to such a system. Boundary conditions are not provided so if the existence is dependent on very stringent initial/boundary conditions, that is no problem. I'm not necessarily looking for a method to solve this problem, I merely want more information/reading material about these types of expressions.
Thanks in advance for any help!