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Finite-dimensional global attractors in Banach spaces |
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| Authors: | Alexandre N Carvalho José A Langa James C Robinson |
| Institution: | a Instituto de Ciências Matemáticas e de Computaçao, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil b Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41080 Sevilla, Spain c Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK |
| Abstract: | We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional. |
| Keywords: | Global attractors Negatively invariant sets Box-counting dimension Banach-Mazur distance |
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