  In 1957, Mullins proposed surface diffusion motion as a model for thermal
grooving. By adopting a small slope approximation, he reduced the model to the
Mullins' linear surface diffusion equation,
  \begin{equation} \nonumber
  ({\rm{ME}})\quad\quad y_t + B y_{xxxx}=0,
  \end{equation} known also more simply as the Mullins' equation. Mullins
sought self-similar solutions to (ME) for planar initial conditions,
prescribing boundary conditions at the thermal groove, as well as far field
decay. He found explicit series solutions which are routinely used in analyzing
thermal grooving to this day.
  While (ME) and the small slope approximation are physically reasonable,
Mullins' choice of boundary conditions is not always appropriate. Here we
present an in depth study of self-similar solutions to the Mullins' equation
for general self-similar boundary conditions, explicitly identifying four
linearly independent solutions defined on $\mathbb{R}\setminus\{0\}$; among
these four solutions, two exhibit unbounded growth and two exhibit asymptotic
decay, far from the origin. We indicate how the full set of solutions can be
used in analyzing the effective boundary conditions from experimental profiles
and in evaluating the governing physical parameters.
