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We consider the time-dependent Ginzburg-Landau model of superconductivity inthe presence of an electric current flowing through a two-dimensional wire. Weshow that when the current is sufficiently strong the solution converges in thelong-time limit to the normal state. We provide two types of upper bounds forthe critical current where such global stability is achieved: by using theprincipal eigenvalue of the magnetic Laplacian associated with the normalmagnetic field, and through the norm of the resolvent of the linearizedsteady-state operator. In the latter case we estimate the resolvent norm inlarge domains by the norms of approximate operators defined on the plane andthe half-plane. We also obtain a lower bound, in large domains, for the abovecritical current by obtaining the current for which the normal state looses itslocal stability.

Global stability of the normal state of superconductors in the presence of a strong electric current