Long-time asymptotics for evolutionary crystal dislocation models

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We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order 2s∈(0,2) acting in one space dimension and the reaction is determined by a 1-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of N⩾2 equally oriented dislocations of size 1. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When s∈(1/2,1), these solutions are shown to be asymptotically stable with respect to odd perturbations.