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Finite-dimensional global attractors in Banach spaces. (English) Zbl 1215.35038

Let \(X\) be a Banach space and \(K\) a compact subset of \(X\). \(N_X(K,\varepsilon)\) is defined as the minimum number of balls in \(X\) of radius \(\varepsilon\) needed to cover \(K\). The upper box-counting dimension \(\dim_B(K)\) of \(K\) is defined by
\[ \dim_B(K)=\limsup_{\varepsilon\to 0} N_X(K,\varepsilon) \frac{\log N_X(K,\varepsilon)}{-\log\varepsilon}. \]
The authors provide bounds on the (upper) box-counting dimension of negatively invariant subsets of Banach spaces. This problem 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 it is shown that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces such as damped wave equation with critical exponent, dissipative parabolic equations in \(L^p(\Omega)\), \(W^{1,p}(\Omega)\)) are finite dimensional.

MSC:

35B41 Attractors
35K90 Abstract parabolic equations
35K58 Semilinear parabolic equations
35L71 Second-order semilinear hyperbolic equations
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