Asymptotics for the Time Dependent Ginzburg–Landau Equations

In this paper, we prove that as →0 the solution of the complex Ginzburg– Landau equationut−Δu=(1/ 2) u(1−u2), inΩ×R+converges to the unique solution of the heat flow for harmonic maps intoS1under the assumption that the initial and boundary maps have zero degree.