Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesuge integrable. Consider the hyperbolic PDE $$ \begin{cases} \partial_{x,y}u & = A\cdot u\\ u(x,\cdot) & = c_1\\ u(\cdot,y) &= c_2 \end{cases} $$
Is there a known closed-form expression for the solution to this type of PDE?
Disclaimer: *This is a question which came up during some research, but since I don't to PDEs and this question seemed basic I did not ask on MO. *