  The diffuse-interface model (DIM) is a widely used tool for modeling fluid
phenomena involving interfaces -- such as, for example, sessile drops (liquid
drops on a solid substrate, surrounded by saturated vapor) and liquid ridges
(two-dimensional sessile drops). In this work, it is proved that, surprisingly,
the DIM does not admit solutions describing static liquid ridges. If, however,
the vapor-to-liquid density ratio is small -- as, for example, for water at
room temperature -- the ridges can still be observed as quasi-static states, as
their evolution is too slow to be distinguishable from evaporation.
Interestingly, the nonexistence theorem cannot be extended to axisymmetric
sessile drops and ridges near a vertical wall, which are not ruled out.
