Periodic regime of Frank-Read source for a dislocation phase field model

We study a two-dimensional phase field model describing the motion of dislocations for a Frank-Read source when an applied shear stress produces a driving rate $ \sigma $. The level sets of the phase describe the dislocation curves and allow their change of topology. The phase solves a nonlinear reaction-diffusion equation of Allen-Cahn type, on a large bounded domain with Neumann boundary conditions. The large domain mimics the whole plane with two holes to which some of the dislocation curves are attached. We show that dislocations are created at the rate $ \omega $ which is a monotone function of the driving rate $ \sigma $. Using the time-shifting method, we study the long-time behavior of the solution, and this allows us to identify a periodic regime whose periodicity is $ 1/\omega $. Moreover, using a semi-norm method, we show that the difference between the solution and the periodic regime converges to zero with an exponential decay in time.