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Multiparameter semigroups and attractors of reaction-diffusion equations in ${\mathbb R}^n$


Author: S. V. Zelik
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 65 (2004).
Journal: Trans. Moscow Math. Soc. 2004, 105-160
MSC (2000): Primary 35B40, 37B40, 37L05
DOI: https://doi.org/10.1090/S0077-1554-04-00145-1
Published electronically: September 30, 2004
MathSciNet review: 2193438
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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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Additional Information

S. V. Zelik
Affiliation: University of Stuttgart, Germany
Email: zelik@mathematik.uni-stuttgart.de

DOI: https://doi.org/10.1090/S0077-1554-04-00145-1
Keywords: Reaction-diffusion equations, multiparameter semigroups, topological entropy, space-time chaos
Published electronically: September 30, 2004
Additional Notes: This research was carried out with the partial support of the INTAS grant no. 00-899 and the CRDF grant no. 2343.
Article copyright: © Copyright 2004 American Mathematical Society

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