# American Institute of Mathematical Sciences

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

 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 $(H_{0}^{1} (Ω))^{N}$.

Citation: Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regularity of global attractors for reaction-diffusion systems with no more than quadratic growth. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] J. L. Lions and E. Magenes, Problémes Aux Limites Non-homogénes et Applications Dunod, Paris, 1968. [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. [13] G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer, 2002. [14] J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New-York, 1983. [15] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. [16] M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Dordrecht, 1988. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] J. L. Lions and E. Magenes, Problémes Aux Limites Non-homogénes et Applications Dunod, Paris, 1968. [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. [13] G. R. Sell and Y. You, Dynamics of Evolutionary Equations Springer, 2002. [14] J. Smoller, Shock Waves and Reaction-Diffusion Equations Springer, New-York, 1983. [15] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. [16] M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers, Dordrecht, 1988. [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. [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.
 [1] Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 [2] Hua Nie, Sze-Bi Hsu, Feng-Bin Wang. Global dynamics of a reaction-diffusion system with intraguild predation and internal storage. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 877-901. doi: 10.3934/dcdsb.2019194 [3] Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 [4] Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103 [5] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [6] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [7] Sebastian Aniţa, William Edward Fitzgibbon, Michel Langlais. Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 805-822. doi: 10.3934/dcdsb.2009.11.805 [8] Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 [9] B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077 [10] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155 [11] Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169 [12] Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 [13] Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 [14] José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299 [15] W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893 [16] Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 [17] Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673 [18] Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817 [19] José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure and Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 [20] Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

2020 Impact Factor: 1.327