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Self-Similar Grooving Solutions to the Mullins' Equation

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.