I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen answered. The PDE $u_t=u_{xx}-u_{yy}$ (or, equivalently, $u_t=u_{xy}$), is the simplest linear second-order PDE that is not elliptic, parabolic, or hyperbolic. Its time-invariant solutions are solutions to the one-dimensional wave equation.

My question is,

How does this PDE compare to the standard parabolic and hyperbolic PDE's in the following categories:

Fundamental solution: is there an integral solution which is a convolution with a fundamental solution?

Smoothness: are solutions infinitely smooth or does the smoothness depend on boundary conditions?

Propagation speed: infinite or finite?

I only recently read Evan's PDE book, and the only question I really took a crack at was the first. I looked for a solution involving exponentials in t, but I couldn't find one. I am interested in this question purely from a classification standpoint. Thanks!

**Edit:** As the comments below indicate, there can be no general solution with arbitrary initial conditions analogous to that for parabolic or hyperbolic equations. Also, solutions need not be infinitely smooth. Since this answer was pieced together from the comments, I'm making this community wiki, in case anyone would like to add to it later.