**MSC:**- 35K55 Nonlinear PDE of parabolic type
- 35B40 Asymptotic behavior of solutions
- 35B65 Smoothness/regularity of solutions of PDE

asymptotic behavior of the solutions of the time-dependent

Ginzburg-Landau equations of superconductivity, in the case where

the applied magnetic field $\H$ is time-dependent. We first prove

existence and uniqueness of solutions with $H^1$-initial data.

This result is obtained under the ``$\phi=-\omega(\nabla\cdot\A)$''

gauge with $\omega>0$. These solutions become then uniformly

bounded in time for the $H^1$-norm, by assuming time-uniform

boundedness on $\H$ and its time derivative.