We consider long-wave unstable interface models of the type

h_t = - (hⁿ h_xxx)_x - B (h^m h_x)_x

where B > 0, and n and m are constants. One interesting aspect of these equations is that there is a range of long-time behavior.

First of all, one would suspect that the destabilizing second-order term could overwhelm the stabilizing fourth-order term. We present a conjecture concerning when this possible, and give analytical and numerical evidence supporting the conjecture (joint work with Andrea Bertozzi of Duke University).

Also, we have classified a large class of nontrivial steady states of PDEs of the above type. There are three types of bounded steady states: constant; nonconstant periodic and $C^\infty$; and compactly supported with lower regularity. We present the linear stability of the smooth steady states and use a Liapunov function to consider the nonlinear stability of steady states. We use the Liapunov function to consider the energy landscape -- to determine what steady states might have heteroclinic connections between them (joint work with Richard Laugesen of the University of Illinois of Champaign-Urbana).