Asymptotic behavior of a nonisothermal Ginzburg-Landau model
Maurizio Grasselli, Hao Wu, Songmu Zheng · Quarterly of Applied Mathematics · 2008
We analyze a system of nonlinear parabolic equations which describes the evolution of an order parameter f f and the relative temperature v v in a superconducting material occupying a bounded domain Ω ⊂ R n \Omega \subset {\mathbb R}^n , n ≤ 3 n\leq 3 . Here the dependent variable f f is subject to the homogeneous Neumann boundary condition, while v v is equal to a time-dependent Dirichlet datum u ~ \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 u ~ \tilde u . Then, we give sufficient conditions on u ~ \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 u ~ \tilde u .