Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic behavior of a nonisothermal Ginzburg-Landau model

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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Additional Information

Maurizio Grasselli
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Milano 20133, Italy
Email: maurizio.grasselli@polimi.it

Hao Wu
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Email: haowufd@yahoo.com

Songmu Zheng
Affiliation: Institute of Mathematics, Fudan University, Shanghai 200433, China
Email: songmuzheng@yahoo.com

DOI: https://doi.org/10.1090/S0033-569X-08-01115-9
Keywords: Ginzburg-Landau systems, phase-field equations, nonautonomous dynamical systems, global attractors, convergence to equilibria, {\L }ojasiewicz-Simon inequality, exponential attractors.
Received by editor(s): July 15, 2007
Published electronically: October 3, 2008
Additional Notes: Dr. Songmu Zheng is the corresponding author.
Article copyright: © Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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