time-dependent diffusion equation - why is it parabolic?

Question

engineer here, looking for some help!

Studying the classification of PDEs I am confused about the following, probably trivial, problem:

The time-dependent diffusion equation is

$$ \frac{\partial \phi}{\partial t} - \alpha \left(\frac{\partial² \phi}{\partial x²} + \frac{\partial² \phi}{\partial y²}\right) = 0$$

and is considered to be a parabolic PDE.

Is it correct that there are 3 independent variables, $x, y \text{ and } t$? If so, how do I apply the common rules for classifying PDEs using the comparison of the discriminant $B² - AC$ with $0$?

Comparing with the general case of such a PDE (as in wikipedia), I'd get $A=-\alpha$, $B=0$ and $C=-\alpha$ again, which appears to be incorrect. But where is my mistake?

Thanks a lot!

https://en.wikipedia.org/wiki/Partial_differential_equation

Answer 1

You are right. There are 3 independent variables here. So, you cannot apply the discriminant criterion $B^2 - AC$, as that holds only for 2 independent variables.

Thus, you must apply the method given in the Wikipedia page for a general PDE. In this case, taking $x_1 = x, x_2 = y, x_3 = t$, the coefficient matrix of the PDE is - $$ \begin{pmatrix} -\alpha & 0 & 0 \\ 0 & -\alpha & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$

As this has eigenvalues $\lambda = -\alpha, -\alpha, 0$, this is a parabolic differential equation.