Some classes of PDEs

I read somewhere (and I think it was mentioned in class) that all linear PDEs can be categorized into either parabolic, hyperbolic, or elliptic types according to: $B^2-4AC$. How do we determine what the values of $A$, $B$ and $C$ are? And does this only apply to second order and smaller PDEs?

Good question. However an answer is more complicated: among 2-nd order equations there are elliptic, hyperbolic, parabolic but also a lot of equations which are neither (and some of them are rather important). Ditto for higher order equations and the systems.

There is no complete classifications of PDEs and cannot be because any reasonable classification should not be based on how equation looks like but on the reasonable analytic properties it exhibits (which IVP or BVP are well-posed etc).

2D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either \begin{equation} u_{xx}+u_{yy}+\text{l.o.t} =f \label{ell-2} \end{equation} or \begin{equation} u_{xx}-u_{yy}+\text{l.o.t.} =f. \label{hyp-2} \end{equation} Here l.o.t. means "lower order terms". (\ref{ell-2}) are elliptic, (\ref{hyp-2}) are hyperbolic.

What to do if one of the 2-nd derivatives is missing? We get \begin{equation} u_{xx}-cu_{y}+\text{l.o.t.} =f. \label{par-2} \end{equation} with $c\ne 0$ and IVP $u|_{y=0}=g$ is well-posed in the direction of $y>0$ if $c>0$ and in direction $y<0$ if $c<0$. We can dismiss $c=0$ as not-interesting.

However this classification leaves out very important Schrödinger equation \begin{equation} u_{xx} +i c u_y=0 \label{Schr-2} \end{equation} with real $c\ne 0$. For it IVP $u|_{y=0}=g$ is well-posed in both directions $y>0$ and $y<0$ but it lacks many properties of parabolic equations (like maximum principle or mollification).

3D If we consider only 2-nd order equations with constant real coefficients then in appropriate coordinates they will look like either \begin{equation} u_{xx}+u_{yy}+u_{zz}\text{l.o.t} =f \label{ell-3} \end{equation} or \begin{equation} u_{xx}+u_{yy}-u_{zz}+\text{l.o.t.} =f. \label{hyp-3} \end{equation} (\ref{ell-3}) are elliptic, (\ref{hyp-3}) are [i]hyperbolic[/i].

Also we get parabolic equations like \begin{equation} u_{xx}+u_{y}-cu_z+\text{l.o.t.} =f. \label{par-3} \end{equation} What about \begin{equation} u_{xx}-u_{yy}-cu_z+\text{l.o.t.} =f? \label{crap-3} \end{equation} Algebraist-formalist would call it parabolic-hyperbolic but since this equation exhibits no interesting analytic properties (unless one considers lack of such properties interesting) it would be a perversion.

Yes, there will be Schrödinger equation \begin{equation} u_{xx} +u_{yy}+i c u_z=0 \label{Schr-3} \end{equation} with real $c\ne 0$ but $u_{xx} -u_{yy}+i c u_z=0$ would also have IVP $u|_{z=0}=g$ well posed in both directions.

4D] Here we would get also elliptic \begin{equation} u_{xx}+u_{yy}+u_{zz}+u_{tt}+\text{l.o.t.} =f, \label{ell-4} \end{equation} hyperbolic \begin{equation} u_{xx}+u_{yy}+u_{zz}-u_{tt}+\text{l.o.t.} =f, \label{hyp-4} \end{equation} but also ultrahyperbolic \begin{equation} u_{xx}+u_{yy}-u_{zz}-u_{tt}+\text{l.o.t.} =f \label{uhyp-4} \end{equation} which exhibits some interesting analytic properties but these equations are way less important than elliptic, hyperbolic or parabolic.

Parabolic and Schrödinger will be here as well.

The notions of elliptic, hyperbolic or parabolic equations are generalized to higher-order equations but most of the randomly written equations do not belong to any of these types and there is no reason to classify them.

To make things even more complicated there are equations changing types from point to point, f.e. Tricomi equation \begin{equation} u_{xx}+xu_{yy}=0 \label{Tric} \end{equation} which is elliptic as $x>0$ and hyperbolic as $x<0$ and at $x=0$ has a "parabolic degeneration". It is a toy-model describing stationary transsonic flow of gas.

My purpose was not to give exact definitions but to explain a situation.