Category:Hyperbolic

A second order linear partial differential equation can be written as $\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0$.
If the matrix $A=(a_{ij})_{ij}$ is indefinite such that $n-1$ eigenvalues have the same sign and the remaining eigenvalue has the other sign, the partial differential equation is called hyperbolic.
An example is the wave equation: $\frac{\partial^2 u}{\partial t^2}-\Delta u = f$, where $\Delta$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ is given.