# Long-time asymptotics for evolutionary crystal dislocation models

## TL;DR

This paper studies the long-term behavior of solutions to generalized crystal dislocation models involving fractional diffusion, revealing stable dislocation configurations and their dynamics over time.

## Contribution

It introduces a family of fractional reaction-diffusion models for dislocations and analyzes their long-time asymptotics and stability properties.

## Key findings

- Dislocations propagate according to a repulsive dynamical system.
- Solutions are asymptotically stable for certain fractional orders.
- The model generalizes classical dislocation equations with fractional operators.

## Abstract

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 $2 s \in (0, 2)$ 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 $N \ge 2$ equally oriented dislocations of size $1$. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When $s \in (1/2, 1)$, these solutions are shown to be asymptotically stable with respect to odd perturbations.