Asymptotic behavior of a nonisothermal Ginzburg-Landau model

Grasselli, Maurizio; Wu, Hao; Zheng, Songmu

doi:10.1090/S0033-569X-08-01115-9

Authors: Maurizio Grasselli, Hao Wu and Songmu Zheng
Journal: Quart. Appl. Math. 66 (2008), 743-770
MSC (2000): Primary 35K55; Secondary 35B40, 35B41, 80A22, 82D55
DOI: https://doi.org/10.1090/S0033-569X-08-01115-9
Published electronically: October 3, 2008
MathSciNet review: 2465143
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Abstract: 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 $\tilde u$.

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