  In this article we provide numerical and analytical evidence that some
degenerate dispersive partial differential equations are ill-posed.
Specifically we study the K(2,2) equation $u_t = (u^2)_{xxx} + (u^2)_{x}$ and
the "degenerate Airy" equation $u_t = 2 u u_{xxx}$. For K(2,2) our results are
computational in nature: we conduct a series of numerical simulations which
demonstrate that data which is very small in $H^2$ can be of unit size at a
fixed time which is independent of the data's size. For the degenerate Airy
equation, our results are fully rigorous: we prove the existence of a compactly
supported self-similar solution which, when combined with certain scaling
invariances, implies ill-posedness (also in $H^2$).
