Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth

Oleksiy V. Kapustyan; Pavlo O. Kasyanov; José Valero

doi:10.3934/dcdsb.2017113

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2017, Volume 22, Issue 5: 1899-1908. Doi: 10.3934/dcdsb.2017113

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Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth

1.
Taras Shevchenko National University of Kyiv, Volodymyrska Street 60,01601, Kyiv, Ukraine
2.
Institute for Applied System Analysis, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" Peremogy ave. 37, Build 35,03056, Kyiv, Ukraine
3.
Universidad Miguel Hernandez de Elche, Centro de Investigación Operativa, Avda. Universidad s/n 03202-Elche (Alicante), Spain

* Corresponding author
* Corresponding author

Received: September 30, 2015

Revised: March 31, 2016

Published: August 2017

The first two authors have been partially supported by the Ukrainian State Fund for Fundamental Researches and the National Academy of Sciences of Ukraine, projects GP/F49/070, r.n. 0113U006191, and F2273/13, r.n. 0113U002978. The third author has been partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2012-31698, and by Junta de Andaluc´ıa under Proyecto de Excelencia P12-FQM-1492.

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Abstract

We consider reaction-diffusion systems in a three-dimensional bounded domain under standard dissipativity conditions and quadratic growth conditions. No smoothness or monotonicity conditions are assumed. We prove that every weak solution is regular and use this fact to show that the global attractor of the corresponding multi-valued semiflow is compact in the space $(H_{0}^{1} (Ω))^{N}$.

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Mathematics Subject Classification: Primary:35B40, 35B41, 35K55, 35Q30, 35Q35;Secondary:37B25, 58C06.

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