TITLE: PDEs of higher order than three?
QUESTION [7 upvotes]: Motivation for question: I know just a little about the general theory of PDEs. I'm working on a project which happens to need examples of PDEs like Laplace's equation. The next step is to look at higher order PDEs in my investigation. I own a few PDE texts and I had hoped to find further examples, but I've noticed that the bulk of my PDE books concern mainly second order PDEs over various dimensional spaces (for example, $n=2$: $u_{xx}+u_{yy}=0$, or $n=3$: $u_{xx}+u_{yy}=u_{tt}$). 
My question is this: do higher order equations (like $u_{xxx}=u_{tt}$, to give a silly example) have a well-developed theory? 
In the case of ODEs I'm aware that we can reduce to a system of first order equations thus solutions to the $n^{th}$-order problem are furnished by the first-order theory. 
Is there a similar story for PDEs? Or, is it just that higher-order PDEs are less interesting?
Thanks in advance for any insights!

REPLY [5 votes]: The first example that comes to mind is the clamped plate equation, which is of 4th order. To see how engineers approach it, click here. More generally, any time your model calls for prescribed values and prescribed derivative on the boundary, you'll probably be dealing with a PDE of order above $2$. For second-order equations such boundary value problems are overdetermined.
There is also a general method for the Dirichlet problem for elliptic operators of any order, namely one can use Gårding's inequality to verify the coercitivity assumption in the Lax-Milgram theorem.   
On the other hand, a recent paper by one of my colleagues  opens with a sobering statement 

There is nothing in the theory of linear strongly elliptic differential operators [of higher orders] that can be called a general existence theorem for solutions to the Neumann problem.

On the positive side, the Fourier transform goes a long way for linear PDE of all orders, especially as far as the regularity of solutions is concerned. The very size of Hörmander's 4-volume treatise "The analysis of linear partial differential operators" suggests that there is something well-developed in there. 
One of the things that make second order equations easier to deal with is the fact that the calculus test for maxima/minima is the second derivative test. The maximum principle is a workhorse in the  theory of elliptic equations of 2nd order. In contrast, the Hadamard maximum principle for biharmonic equation does not go very far (see page 2 of the paper).