Comptes Rendus
no. 2
Partial Differential Equations
On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors
[Sur le comportement en temps grand des solutions d'équations paraboliques d'ordre quatre, et l'entropie de leurs attracteurs]

Présenté par : Haïm Brezis

M.A. Efendiev1 ; L.A. Peletier2, 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.

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.

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.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2006.10.028
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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     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},
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     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/

[1] G.T. Dee; W. van Saarloos Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., Volume 60 (1988), pp. 2641-2644

[2] M.A. Efendiev; S. Zelik The attractor for a nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math., Volume LIV (2001), pp. 625-688

[3] M.A. Efendiev; S. Zelik Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain, J. Dynam. Differential Equations, Volume 14 (2002), pp. 369-404

[4] L.A. Peletier; V. Rottschäfer Pattern selection of solutions of the Swift–Hohenberg equation, Physica D, Volume 194 (2004), pp. 95-126

[5] L.A. Peletier; W.C. Troy Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001

[6] J.B. Swift; P.C. Hohenberg Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, Volume 15 (1977), pp. 319-328

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