Multiparameter semigroups and attractors of reaction-diffusion equations in ℝⁿ

Zelik, S.

doi:10.1090/S0077-1554-04-00145-1

Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$
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by S. V. Zelik
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2004, 105-160

DOI: https://doi.org/10.1090/S0077-1554-04-00145-1

Published electronically: September 30, 2004

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Abstract:

The space-time dynamics generated by a system of reaction-diffusion equations in $\mathbb R^n$ on its global attractor are studied in this paper. To describe these dynamics the extended $(n+1)$-parameter semigroup generated by the solution operator of the system and the $n$-parameter group of spatial translations is introduced and their dynamic properties are studied. In particular, several new dynamic characteristics of the action of this semigroup on the attractor are constructed, generalizing the notions of fractal dimension and topological entropy, and relations between them are studied. Moreover, under certain natural conditions a description of the dynamics is obtained in terms of homeomorphic embeddings of multidimensional Bernoulli schemes with infinitely many symbols.

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