Second order hyperbolic equation

Question

Consider $\Omega$ a regular domain of $\mathbb{R}^2$. I'm looking for criterion to ensure the well posedness (or not) of this hyperbolic equation: find $u\in H^1_0(\Omega)$ such that $$div(A\nabla u)=f\in\mathbb{L}^2(\Omega)$$ in which $A$ is $2\times2$ symmetric matrix with eigenvalues $a_1$ and $b_1$ such that $a_1b_1<0$.