The particular PDE I would like to know about would be
$$\partial_t u = D(\partial^2_{x} +\partial^2_y) u + AS((\partial_x)^2+(\partial_y)^2)u +AA \partial_x\partial_yu + c(1-c)$$
The classification of Partial Differential Equations into elliptic, parabolic and hyperbolic is formally defined for linear second order PDEs only.
Then, assuming $u=u(x,y)$, you can determine its type by bringing it to its canonical form
$$ \mathsf{A}\,u_{xx} + 2\,\mathsf{B}\,u_{xy} + \mathsf{C}\,u_{yy} +
\mathsf{D}\, u_{x} + \mathsf{E} \, u_{y} + \mathsf{F} = 0 $$
In that case, if
- $\mathsf{B}^2 - 4\mathsf{A}\mathsf{C} <0$, then the equation is elliptic,
- $\mathsf{B}^2 - 4\mathsf{A}\mathsf{C} =0$, then the equation is parabolic,
- $\mathsf{B}^2 - 4\mathsf{A}\mathsf{C} >0$, then the equation is hyperbolic.
One way to apply this classification to a general (e.g. quasilinear, semilinear, nonlinear) second order PDE is to linearize it. It is actually unclear whether your original PDE is linear or not:
If $A$, $D$, and $S$ are constants, or functions which depend on $(x,y,t)$, then the equation is linear,
so you can write out the canonical form and determine the type of PDE.
If, however, $A$, $D$, and $S$ are (nonlinear) functions, which depend on $u$ or its derivatives, then
you might have to compute a linearization of PDE first, and then determine its type.
Note that for the classification purposes we do not make a distinction between spatial ($x,y$) and temporal ($t$) variables.
In your case function $ u = u(x,y,t) $ depends, apparently, on three variables.
This means that you will have to use generalized classification for PDEs defined on $\mathbb{R}^{3}$, which can be found in the Wikipedia article about PDEs.
By your notation I will assume you are looking at operators of the form
$$ F(D)u = \frac12\sum_{i,j=1}^n H_{ij}\partial_{x_ix_j}u + \sum_{i=1}^n g_i\partial_{x_i}u + cu = 0. $$
First question: is that interpretation correct?
The classification given in Strauss is for PDEs in two dimensions only; in higher dimensions you still have a classification, but it's no longer exhaustive. Also you implicitly assume $H \neq 0$ so the system is genuinely of second order, which is why the parabolic case assumes exactly one eigenvalue is zero.
In general dimensions this is somewhat convention-dependent, but you have the following classification:
- The equation is elliptic if $H$ is non-singular and all eigenvalues have the same sign.
- The equation is hyperbolic if $H$ is non-singular and all but one eigenvalue has the same sign.
- The equation is parabolic if $H$ has one zero eigenvalue, and all other eigenvalues have the same sign. This is somewhat convention-dependent however, and one usually assumes extra structure in this case.
Note this is not a complete classification; if $H$ is non-singular but neither elliptic nor hyperbolic, then it is sometimes referred to as ultrahyperbolic - however this is limited literature on equations of these type.
Generally you should think of the model equations: Laplace's equation, the heat equation and the wave equation. The classification also extends more generally to nonlinear equations, etc, and while the convention varies this is generally based on how solutions to these equations behave.
For the parabolic and hyperbolic case, if $H$ is constant we see we have a distinguished direction $v$ - in the parabolic case this corresponds to the zero eigenvector, while in the hyperbolic case this corresponds to the eigenvector whose eigenvalue has different sign. By a change of variables we can assume $v = e_n,$ and that writing $x_n=t$ the equation takes the form
$$ g_n \partial_{t}u + \frac12\sum_{i,j=1}^{n-1} H_{ij}\partial_{x_ix_j}u + \sum_{i=1}^{n-1} g_i \partial_{x_i} u + cu = 0 $$
in the parabolic case, and
$$ \frac12H_{nn}\partial_{t}^2u + \frac12\sum_{i,j=1}^{n-1} H_{ij}\partial_{x_ix_j}u + \sum_{i=1}^{n-1} g_i \partial_{x_i} u +g_n\partial_tu+ cu = 0 $$
in the hyperbolic case (I am omitting the details here, but some linear algebra is involved). Therefore we see we naturally obtain a distinguished direction, which we suggestively denote as $t$ - which was completely determined by the equation! Note in the parabolic case we usually assume $g_n \neq 0,$ as otherwise the equation reduces to an elliptic equation in the first $(n-1)$ variables, and in the hyperbolic case $H_{nn} < 0$ necessarily because $H$ is non-degenerate (assuming the other eigenvalues are positive).
Generally when working with parabolic and hyperbolic equations we assume there is a distinguished direction $\partial_t,$ even for more general variable coefficient and nonlinear equations${}^\ast$ - this is because in the applications we are interested in there is a natural time direction, and it is more convenient for purposes of analysis.
${}^\ast$Note the classification is also used to distinguish nonlinear PDEs, but here there is no standard way of doing this. Typically we require a suitable linearised equation to be elliptic/parabolic/hyperbolic, but what this means depends on the particular application. Also in this case we don't always have a distinguished time direction, which notably arises in the context of general relativity.
Now that I've explained how a time direction naturally arises in the parabolic and hyperbolic cases, I can turn to your question about propagation of information.
What exactly is meant by "information propagation"? Without necessarily including a time variable, can I assume that this is some bound on $\max_i|\partial u_i/\partial u_j|$?
Propagation of information generally refers to how local changes in the initial/boundary data is reflected in the corresponding solution. For this however, note that the type of boundary conditions varies depending on the equations we consider.
For elliptic equations we consider boundary value problems, where for a bounded domain $\Omega \subset \Bbb R^n$ we seek solutions $u$ subject to conditions on the boundary; e.g. $u = g$ on $\partial\Omega$ (Dirichlet) or $\partial_{\nu}u = h$ on $\partial\Omega$ where $\partial_{\nu}$ is the normal derivative (Neumann).
For parabolic and hyperbolic problems we consider initial boundary value problems, typically on domains of the form $\Omega' \times (0,T)$ where $\Omega' \subset \Bbb R^{n-1}$ is a bounded domain. In both cases we prescribe $u$ on $\partial\Omega' \times (0,T)$ (the boundary part) and in the parabolic case $u$ on $\Omega' \times \{0\}$ also (the initial part). For hyperbolic problems we need to prescribe initial values for both $u$ and $\partial_tu$ at $t=0.$
The reason we impose these conditions is because it turns out this is necessary to ensure the equation is well-posed; if you prescribe suitably regular initial/boundary data, then you can show a unique solution exists. There are many examples that show well-posedness fails if you don't impose the correct conditions, and for instance this explains why there is no time direction if your equation is elliptic.
In the elliptic and parabolic cases, local changes to initial/boundary data can change the solution at every point, which is what we mean by infinite speed of propagation. For wave equations this is not the case however, and changes to initial data is propagates along a 'wave cone.'
The details of this are lengthy, so I refer you to Chapter 2 of Strauss' text which discusses these phenomena in the context of the wave and diffusion (heat) equations.
Why would some systems result in finite information propagation rate and others necessarily have infinite information propagation rate?
This is a rather deep question, but as I've alluded to in the above discussion the behaviour of solutions to PDEs greatly varies depending on their classification. This is why for instance there is no unified theory of PDEs - small changes to the equation can result is very different behavior, which is to be expected because different equations model different physical phenomena.
Best Answer
We are considering the one-dimensional conservation law $$ \frac{\partial\mathbf{U}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{U})}{\partial x} =\mathbf{0}\quad\text{where}\quad\mathbf{U}\equiv\begin{bmatrix} \rho\\ v \end{bmatrix} $$ or in quasi-linear form $$ \frac{\partial\mathbf{U}}{\partial t} + \mathbf{A}\frac{\partial\mathbf{U}}{\partial x}=\mathbf{0} \quad\text{where}\quad\mathbf{A}\equiv\frac{\partial\mathbf{F}}{\partial\mathbf{U}}. $$ Following Toro [ Riemann Solvers and Numerical Methods in Fluid Dynamics ] and LeVeque [ Numerical Methods for Conservation Laws ]
"The system is hyperbolic [...], if $\mathbf{A}$ has $m$ real Eigenvalues $\lambda_1,...,\lambda_m$ […]. The system is said to be strictly hyperbolic if the eigenvalues $\lambda_i$ are all distinct."
and
"The system is said to be elliptic […], if none of the eigenvalues $\lambda_i$ of $\mathbf{A}$ are real [thus all eigenvalues $\lambda_i$ need to be complex]."
2. In particular what type of system will it be if it has two real but repeated eigenvalues?
The eigenvalues $\lambda_i$ are real, but not distinct - the system is hyperbolic, even tough it is not strictly hyperbolic. Therefor $\mathbf{A}$ is only diagonizable, if $m$ linear independent eigenvectors exist.
1. Is there an analogous criteria to determine whether the system is elliptic or parabolic?
If all eigenvalues are complex the system is elliptic. Unfortunately for a parabolic system, I am not aware of a analogous criteria; From a physical point of view, a parabolic equation corresponds to a second derivative in space (e.g. transient head conduction). Therefor the flux $\mathbf{F}$ is not only dependent in $\mathbf{U}$, but also in its gradient; $$ \mathbf{F}(\mathbf{U},\nabla\mathbf{U}) $$ and the quasi-linear form (which is the basis of our definition) does not hold anymore. If someone has more insight for this case, please leave a comment.