Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$.
By using the separation of variables method, we find that the exact solution $u$ is given by $$ u(x, y) = \sum_{n, m \geq 1} \frac{f_{nm}}{\left(\frac{an\pi}{L_x}\right)^2 - \left(\frac{bm\pi}{L_y}\right)^2} \sin\left(\frac{n\pi x}{L_x}\right) \sin\left(\frac{m\pi y}{L_y}\right), $$ where $f_{nm}$ are the Fourier coefficients of $f$. This solution exists if $$ \left|\frac{an\pi}{L_x} - \frac{bm\pi}{L_y}\right| \geq C $$ for all $n, m \geq 1$. However, it may not be possible to satisfy this condition, as any irrational number can be approximated by rational numbers (by Diophantine approximation). Am I correct?
Also, we can observe that if one of $a$ or $b$ is zero the equation reduces to the Laplace equation in 1D. In this case, the system is well-posed.
Is there any optimal condition on $a$ and $b$ so that it is well-posed? in which space?. Are there any references to this type of equation?
I appreciate any help you can provide.
