  With few exceptions, known explicit solutions of the curve shortening flow
(CSE) of a plane curve, can be constructed by classical Lie point symmetry
reductions or by functional separation of variables. One of the functionally
separated solutions is the exact curve shortening flow of a closed, convex
"oval"-shaped curve and another is the smoothing of an initial periodic curve
that is close to a square wave. The types of anisotropic evaporation
coefficient are found for which the evaporation-condensation evolution does or
does not have solutions that are analogous to the basic solutions of the CSE,
namely the grim reaper travelling wave, the homothetic shrinking closed curve
and the homothetically expanding grain boundary groove. Using equivalence
classes of anisotropic diffusion equations, it is shown that physical models of
evaporation-condensation must have a diffusivity function that decreases as the
inverse square of large slope. Some exact separated solutions are constructed
for physically consistent anisotropic diffusion equations.
