Following on his dissertation, Yonghoi Koo collaborated with me to show that if two pieces of a hypersurface, flowing by mean curvature plus a forcing term, touch at time 0 without crossing thereafter, then the viscosity solution has size of order the one-third power of time. That is, fattening of the level sets in this situation happens much more rapidly than would be expected with parabolic scaling. See the Postscript version or the PDF version of our joint paper [42]. This non-self-similar scaling is related to the non-conical shape of their common exterior at time 0.

[42]. Sharp Growth Rate for Generalized Solutions Evolving by Mean Curvature Plus a Forcing Term (with Yonghoi Koo), Journal für die Reine und Angewandte Mathematik 538, 1-24 (2001). Postscript version or PDF version