Evolution Equations and Control Theory (EECT)

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

Pages: 345 - 356, Volume 6, Issue 3, September 2017      doi:10.3934/eect.2017018

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María Anguiano - Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, P. O. Box 1160, 41080-Sevilla, Spain (email)
Alain Haraux - CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France (email)

Abstract: 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.

Keywords:  Fractal dimension, attractors, entropy.
Mathematics Subject Classification:  37L30, 35B41.

Received: April 2017;      Revised: May 2017;      Available Online: July 2017.