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 & 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. GorbanO. 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. KapustyanP. 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. KapustyanP. 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. KapustyanP. 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. KasyanovL. 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. VishikS. 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. GorbanO. 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. KapustyanP. 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. KapustyanP. 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. KapustyanP. 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. KasyanovL. 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. VishikS. 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

[1]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[2]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049

[3]

Mustapha Yebdri. Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021036

[4]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[5]

M. Syed Ali, L. Palanisamy, Nallappan Gunasekaran, Ahmed Alsaedi, Bashir Ahmad. Finite-time exponential synchronization of reaction-diffusion delayed complex-dynamical networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1465-1477. doi: 10.3934/dcdss.2020395

[6]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024

[7]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405

[8]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003

[9]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400

[10]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[11]

Masaharu Taniguchi. Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3981-3995. doi: 10.3934/dcds.2020126

[12]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[13]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053

[14]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[15]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[16]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[17]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[18]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[19]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[20]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329

2019 Impact Factor: 1.27

Article outline

[Back to Top]