  The nonequilibrium process in dislocation dynamics and its relaxation to the
metastable transition profile is crucial for understanding the plastic
deformation caused by line defects in materials. In this paper, we consider the
full dynamics of a scalar dislocation model in two dimensions described by the
bulk diffusion equation coupled with dynamic boundary condition on the
interface, where a nonconvex misfit potential, due to the presence of
dislocation, yields an interfacial reaction term on the interface. We prove the
dynamic solution to this bulk-interface coupled system will uniformly converge
to the metastable transition profile, which has a bi-states with fat-tail decay
rate at the far fields. This global stability for the metastable pattern is the
first result for a bulk-interface coupled dynamics driven only by an
interfacial reaction on the slip plane.
