Classification of this nonlinear PDE into elliptic, hyperbolic, etc. I wanted to know how one would classify a nonlinear PDE into elliptic, hyperbolic or parabolic forms.
The particular PDE I would like to know about would be
\begin{align}
  \partial_t u &= D(\partial^2_{x} +\partial^2_y) u + AS((\partial_x u)^2+(\partial_y u)^2) +AA (\partial_xu)(\partial_yu) + c(1-c)
\end{align}
 A: 
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. 
