Can one classify nonlinear PDEs? I want to classify a PDE into (hyperbolic, elliptic, parabolic) which I know how to do when I am confronted with a linear PDE. 

Can one classify nonlinear PDEs? If yes, how?


I have the following PDE which I want to classify: $$(u_{xx})^2+\exp{(y)}+(u_{yy})=\sin{(x)}$$
Another example is the following:$$(x^2-1)u_{xx}+2xyu_{xx}+(y^2-1)u_{yy}=xu_{x}+yu_{y}$$
with $u:(x,y)\rightarrow u(x,y)$.
The second example is linear. 
 A: Nonlinear PDEs do not always admit a nice classification. For your PDE
$$u_{xx}^2 + u_{yy} = f(x,y)$$
the classification depends on the sign of $u_{xx}$. If $u_{xx}>0$ the equation is elliptic. If $u_{xx}<0$ it is hyperbolic. If $u_{xx}$ switches sign, then the equation can be elliptic in some regions of the domain, and hyperbolic in other regions. This is one reason why nonlinear PDE are more interesting.
A classic example is the Monge-Ampere equation
$$u_{xx}u_{yy} - u_{xy}^2 = f(x,y),$$
which shares some similarities with your PDE. When $u$ is convex, the Monge-Ampere equation is elliptic. This is why people sometimes refer to the "elliptic Monge-Ampere" equation to mean they are studying convex solutions. 
There is a nice definition of elliptic for nonlinear PDEs. A PDE 
$$F(\nabla^2u,\nabla u, u,x) = 0$$
is called elliptic if
$$F(X,p,z,x) \geq F(Y,p,z,x)$$
whenever $X \leq Y$. Here, $X$ and $Y$ are symmetric $n\times n$ matrices and $X\leq Y$ means that $Y-X$ is non-negative definite. 
