Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations

Authors: Valeria Berti and Stefania Gatti
Journal: Quart. Appl. Math. 64 (2006), 617-639
MSC (2000): Primary 35B41
DOI: https://doi.org/10.1090/S0033-569X-06-01044-9
Published electronically: October 16, 2006
MathSciNet review: 2284463
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Abstract: This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bidimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.

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Additional Information

Valeria Berti
Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I-40126 Bologna, Italy
Email: berti@dm.unibo.it

Stefania Gatti
Affiliation: Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, I-44100 Ferrara, Italy
Email: s.gatti@economia.unife.it

DOI: https://doi.org/10.1090/S0033-569X-06-01044-9
Keywords: Ginzburg-Landau-Maxwell equations, superconductivity, strongly continuous semigroup, global attractor, exponential attractor
Received by editor(s): April 11, 2005
Published electronically: October 16, 2006
Additional Notes: Research performed under the auspices of G.N.F.M. - I.N.D.A.M. and partially supported by the Italian MIUR PRIN Research Project “Dinamica a lungo termine e problemi di regolarità per modelli di cambiamento di fase" and by University of Bologna - Funds for selected topics.
Article copyright: © Copyright 2006 Brown University

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