Relaxation of the Curve Shortening Flow via the Parabolic Ginzburg-Landau equation

In this paper we study how to find solutions $$u_\epsilon$$ to the parabolic Ginzburg–Landau equation that as $$\epsilon \to 0$$ have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a uniform profile for the solution $$u_\epsilon$$ up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant