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