# Category:Parabolic

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 positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic.
An example is the heat equation: $\frac{\partial u}{\partial t}-\Delta u = f$, where $\Delta$ denotes the Laplace operator, $u$ is the unknown, and the function $f$ is given.