  We consider a singular parabolic equation of form
  \[
  u_t = u_{xx} + \frac{\alpha}{2}(\mathrm{sgn}\,u_x)_x
  \]
  with periodic boundary conditions. Solutions to this kind of equations
exhibit competition between smoothing due to one-dimensional Laplace operator
and tendency to create flat facets due to strongly nonlinear operator
$(\mathrm{sgn}\,u_x)_x$ coming from the total variation flow. We present
results concerning analysis of qualitative behaviour and regularity of the
solutions. Our main result states that locally (between moments when facets
merge), the evolution is described by a system of free boundary problems for
$u$ in intervals between facets coupled with equations of evolution of facets.
In particular, we provide a proper law governing evolution of endpoints of
facets. This leads to local smoothness of the motion of endpoints and the
unfaceted part of the solution.
