Classification of second-order PDEs

Definition

Let $\Omega$ be an open set in $\R^n$ and consider a linear PDE of order $2$ in $\Omega$. $$\sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u + c(x) \, u = f(x).$$ Its coefficients are real and $a_{ij} = a_{ji}$, so that $A(x) = (a_{ij}(x))$ is a real symmetric matrix. At a point $x \in \Omega$ the equation is called:

1. elliptic if all eigenvalues of $A(x)$ are nonzero and have the same sign;
2. hyperbolic if all eigenvalues of $A(x)$ are nonzero, and exactly one of them has a sign opposite to the others;
3. ultrahyperbolic if all eigenvalues are nonzero, and at least two are positive and at least two are negative;
4. parabolic if exactly one eigenvalue of $A(x)$ is zero and the others are nonzero and have the same sign.

The equation has one of these types in $\Omega$ if it has that type at every point of $\Omega$.