Asymptotic behavior of a nonisothermal Ginzburg-Landau model

We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter $f$ and the relative temperature $v$ in a superconducting material occupying a bounded domain $\Omega\subset {\mathbb{R}}^n$, $n\leq 3$. Here the dependent variable $f$ is subject to the homogeneous Neumann boundary condition, while $v$ is equal to a time-dependent Dirichlet datum $\tilde u$ on the boundary. Therefore, the corresponding dynamical system is nonautonomous. Our main goal is to analyze the asymptotic behavior of its solutions. We first show that the system has a bounded absorbing set and a global attractor which are uniform with respect to a sufficiently general class of $\tilde u$. Then, we give sufficient conditions on $\tilde u$ which ensure the convergence of a given trajectory to a single stationary state and we estimate the convergence rate. Finally, we demonstrate the existence of an exponential attractor of finite fractal dimension for quasi-periodic or stabilizing boundary data $\tilde u$.