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Kolmogorov \(\varepsilon\)-entropy estimates for the uniform attractors of non-autonomous reaction-diffusion systems. (English. Russian original) Zbl 0915.35056

Sb. Math. 189, No. 2, 81-110 (1998); translation from Mat. Sb. 189, No. 2, 235-263 (1998).
The Kolmogorov \(\varepsilon\)-entropy of the uniform attractor \({\mathcal A}\) of a family of non-autonomous reaction-diffusion systems with external forces \(g(x,t)\) is studied. The external forces \(g(x,t)\) are assumed to belong to some subset \(\Sigma\) of \(C(\mathbb{R}; H)\), where \(H= (L_2(\Omega))^N\), that is invariant under the group of \(t\)-translations. Furthermore, \(\Sigma\) is compact in \(C(\mathbb{R}; H)\).
An estimate for the \(\varepsilon\)-entropy of the uniform attractor \({\mathcal A}\) is given in terms of the \(\varepsilon_1= \varepsilon_1(\varepsilon)\)-entropy of the compact subset \(\Sigma_{0,l}\) of \(C([0, l]; H)\) consisting of the restrictions of the external forces \(g(x,t)\in\Sigma\) to the interval \([0,l]\), \(l= l(\varepsilon)\) (here \(\varepsilon_1(\varepsilon)\sim \mu\varepsilon\) and \(l(\varepsilon)\sim \tau\log_2(1/\varepsilon)\)). This general estimate is illustrated by several examples from different fields of mathematical physics and information theory.

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

external forces
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