PDE U_t = aU_xy (mixed derivatives)

[RedBranchKnight]

I am trying to solve

(1) U_t = 2bU_xy (as part of U_t = aU_xx + 2bU_xy + cU_yy)

using centred finite difference method. When a > 0 everyhing is OK but when a < 0 I get some oscillation problems.

My questions are:

1. is there a pde theory for (1)?
2. what is the 'motivation' for (1) in terms of information flow to and from boundaries/
3. maybe my fdm schemes destroy ellipticity which is the reason why I get oscillations.
4. These oscillations seem to be related to the initial condition in some way but I have not figured it out

thanks for any help!
RBK