Given known functions $a(x,y)$ and $b(x,y)$, I have a linear, second-order, hyperbolic PDE for a surface $z(x,y)$. The PDE is of the form:

$z_{xx} - a^2 z_{yy} + ab z_x - b z_y = 0$.

The PDE does not have $z_{xy}$ term, $z$ term and constant term.

Along a closed boundary contour, it is known that both $z_x$ and $z_y$ approach infinity. But the limiting ratio $c(x,y) = {z_y}/{z_x}$ is known at the boundary.

The boundary conditions are somewhat unusual, but they arise naturally in my application.

Can the above PDE be solved with such BCs? If I need to specify additional conditions, what should they be? (I suppose at the least I will need to specify the value of z at some points in the domain.)