2-D heat equation being parabolic?

Question

This is a mighty dumb question but I can not seem to figure out how a 2-D heat equation is parabolic. $$ \frac{\partial u}{\partial t}=\Delta u $$ I know that the discriminant : $b^2-ac=0$ for a parabolic PDE.

Here, $b=0,a=c=-1$ . Then how is it parabolic ? I think I am missing out something.

Answer 1

$u$ is a function of $x$ and $t$. In the discriminant, $c$ is the co-efficient of $\frac{\partial^2 u}{\partial t^2}$, which is $0$, so the discriminant is indeed $0$.