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July  2017, 22(5): 1899-1908. doi: 10.3934/dcdsb.2017113

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

Oleksiy V. Kapustyan 1,, , Pavlo O. Kasyanov 2, and  José Valero 3,

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

Received  October 2015 Revised  April 2016 Published  March 2017

Fund Project: 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.

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 $(H_{0}^{1} (Ω))^{N}$.

Keywords: Reaction-diffusion system, multivalued dynamical system, global attractor.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55, 35Q30, 35Q35; Secondary: 37B25, 58C0.
Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1899-1908. doi: 10.3934/dcdsb.2017113
References:
[1]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. Google Scholar

[2]

N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusions in unbounded domains, in Continuous and distributed systems, Solid Mechanics and its Applications, (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing, Switzerland, 211 (2013), 205-220. Google Scholar

[3]

N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar

[4]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[5]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Communications on Pure and Applied Analysis, 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[6]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the uniform global attractor for general non-autonomous reaction-diffusion systems, in Continuous and distributed systems, Solid Mechanics and its Applications 211 (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing Switzerland, 2014,163-180. Google Scholar

[7]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264.   Google Scholar

[8]

O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272.  doi: 10.1016/j.jmaa.2009.04.010.  Google Scholar

[9]

O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos, 20 (2010), 2723-2734.  doi: 10.1142/S0218127410027313.  Google Scholar

[10]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

[11]

J. L. Lions and E. Magenes, Problémes Aux Limites Non-homogénes et Applications Dunod, Paris, 1968. Google Scholar

[12]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[13]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer, 2002. Google Scholar

[14]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New-York, 1983. Google Scholar

[15]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. Google Scholar

[16]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Dordrecht, 1988. Google Scholar

[17]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system, Doklady RAN, 435 (2010), 155-159.  doi: 10.1134/S1064562410060086.  Google Scholar

[18]

M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, in Continuous and distributed systems, Solid Mechanics and its Applications 211 (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing, Switzerland, 2014,149-162. Google Scholar

show all references

References:
[1]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. Google Scholar

[2]

N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusions in unbounded domains, in Continuous and distributed systems, Solid Mechanics and its Applications, (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing, Switzerland, 211 (2013), 205-220. Google Scholar

[3]

N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Anal., 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar

[4]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst., 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar

[5]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Communications on Pure and Applied Analysis, 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[6]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the uniform global attractor for general non-autonomous reaction-diffusion systems, in Continuous and distributed systems, Solid Mechanics and its Applications 211 (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing Switzerland, 2014,163-180. Google Scholar

[7]

O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure of the global attractor for weak solutions of a reaction-diffusion equation, Appl. Math. Inf. Sci., 9 (2015), 2257-2264.   Google Scholar

[8]

O. V. Kapustyan and J. Valero, On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion, J. Math. Anal. Appl., 357 (2009), 254-272.  doi: 10.1016/j.jmaa.2009.04.010.  Google Scholar

[9]

O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Internat. J. Bifur. Chaos, 20 (2010), 2723-2734.  doi: 10.1142/S0218127410027313.  Google Scholar

[10]

P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued Var. Anal., 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar

[11]

J. L. Lions and E. Magenes, Problémes Aux Limites Non-homogénes et Applications Dunod, Paris, 1968. Google Scholar

[12]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[13]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer, 2002. Google Scholar

[14]

J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New-York, 1983. Google Scholar

[15]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. Google Scholar

[16]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Dordrecht, 1988. Google Scholar

[17]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor of dissipative reaction-diffusion system, Doklady RAN, 435 (2010), 155-159.  doi: 10.1134/S1064562410060086.  Google Scholar

[18]

M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, in Continuous and distributed systems, Solid Mechanics and its Applications 211 (M. Z. Zgurovsky and V. A. Sadovnichiy eds. ), Springer International Publishing, Switzerland, 2014,149-162. Google Scholar

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