On the large time behavior of solutions of fourth order parabolic equations and <i>ε</i>-entropy of their attractors

M.A. Efendiev; L.A. Peletier

doi:10.1016/j.crma.2006.10.028

Partial Differential Equations

On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors

Presented by: Haïm Brezis

M.A. Efendiev ¹; L.A. Peletier ^2,³

¹ GSF/Technical University of Münich, Center for Mathematical Sciences, 85747 Garchung-Münich, Germany
² Mathematical Institute, Leiden University, PO Box 9512, NL-2300 RA Leiden, The Netherlands
³ Centrum voor Wiskunde en Informatica (CWI), NL-1090 GB Amsterdam, The Netherlands

Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 93-96.

Abstract
Résumé

We study the large time behavior of solutions of a class of fourth order parabolic equations defined on unbounded domains. Specific examples of the equations we study are the Swift–Hohenberg equation and the Extended Fisher–Kolmogorov equation. We establish the existence of a global attractor in a local topology. Since the dynamics is infinite dimensional, we use the Kolmogorov ε-entropy as a measure, deriving a sharp upper and lower bound.

Nous étudions le comportement pour des grandes valeurs du temps des solutions d'une classe d'équations parabolique d'ordre quatre définie sur des domaines non bornés. Les examples spécifiques que nous considérons sont l'équation de Swift–Hohenberg et une généralisation de l'équation de Fisher–Kolmogorov. Nous démontrons l'existence d'un attracteur global dans une topologie locale, et nous obtenons des limites supérieure et inférieure de l'entropie de Kolmogorov.

Received: 2006-07-06
Published online: 2006-12-08

DOI: 10.1016/j.crma.2006.10.028

Author's affiliations:

M.A. Efendiev ¹; L.A. Peletier ^{2, 3}

¹ GSF/Technical University of Münich, Center for Mathematical Sciences, 85747 Garchung-Münich, Germany
² Mathematical Institute, Leiden University, PO Box 9512, NL-2300 RA Leiden, The Netherlands
³ Centrum voor Wiskunde en Informatica (CWI), NL-1090 GB Amsterdam, The Netherlands

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     author = {M.A. Efendiev and L.A. Peletier},
     title = {On the large time behavior of solutions of fourth order parabolic equations and \protect\emph{\ensuremath{\varepsilon}}-entropy of their attractors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {93--96},
     publisher = {Elsevier},
     volume = {344},
     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2006.10.028},
     language = {en},
}

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M.A. Efendiev; L.A. Peletier. On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors. Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 93-96. doi : 10.1016/j.crma.2006.10.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.028/

References
Cited by

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