Parabolic PDE in higher dimension

Question

I know that a second order linear PDE $$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0$$ is said to be parabolic if $$B^2-AC=0.$$

But this definition only deals with functions of two variables (x,y). How can I say if a PDE in say three variables is parabolic?

Answer 1

The second-order part of the differential equation can be expressed as $\partial_i(a_{ij}\partial_j u)$ for some symmetric matrix $A = \{a_{ij}\}$. Since $A$ is symmetric, it is orthogonally diagonalizable and all of its eigenvalues are real. We use this to rotate the coordinate system such that the second-order part is $\lambda_i\partial_i'^2u$, where $\partial'$ is the derivative with respect to the new coordinates and $\lambda_i$ are the eigenvalues of $A$. We can then classify the equation based on the signs of the $\lambda_i$:

• Elliptic: $\lambda_i$ are all positive or all negative.
• Parabolic: there are some zero $\lambda_i$, but the nonzero $\lambda_i$ are all positive or all negative.
• Hyperbolic: there are both positive and negative $\lambda_i$.