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The $\varepsilon$-entropy of some infinite dimensional compact ellipsoids and fractal dimension of attractors

  • * Corresponding author: Alain Haraux

    * Corresponding author: Alain Haraux
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  • We prove an estimation of the Kolmogorov $\varepsilon$ -entropy in $H$ of the unitary ball in the space $V$ , where $H$ is a Hilbert space and $V$ is a Sobolev-like subspace of $H$ . Then, by means of Zelik's result [7], an estimate of the fractal dimension of the attractors of some nonlinear parabolic equations is established.

    Mathematics Subject Classification: 37L30, 35B41.

    Citation:

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    [3] V. V. Chepyzhov and A. A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Analysis, 44 (2001), 811-819.  doi: 10.1016/S0362-546X(99)00309-0.
    [4] I. DumerM. S. Pinsker and V. V. Prelov, On coverings of ellipsoids in Euclidean spaces, Transactions on Information Theory, 50 (2004), 2348-2356.  doi: 10.1109/TIT.2004.834759.
    [5] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318. 
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    [7] S. Zelik, The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and its dimension, Rend. Accad. Naz. Sci. XL Mem. Mem. Math. Appl., 24 (2000), 1-25. 
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