A non-linear pseudodifferential equation was studied which described the dynamics of dislocations in crystals. The long-term asymptotics of the solutions were  described by self-similar profiles.

Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions. P.Biler, G.Karch, R.Monneau: Communications in Mathematical Physics, 2010, 294[1], 145-68