# American Institute of Mathematical Sciences

July  2017, 22(5): 2053-2065. doi: 10.3934/dcdsb.2017120

## Uniform global attractors for non-autonomous dissipative dynamical systems

 1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Peremogy ave., 37, build, 1,03056, Kyiv, Ukraine 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA 3 Institute for Applied System Analysis, National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Peremogy ave., 37, build, 35,03056, Kyiv, Ukraine

* Corresponding author: Pavlo Kasyanov

Received  January 2016 Revised  February 2016 Published  March 2017

Fund Project: The research was partially supported by the National Academy of Sciences of Ukraine under grant 2284/15 and by Grant of the President of Ukraine GP/F61/017.

In this paper we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.

Citation: Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
##### References:
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Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1975. Google Scholar [12] M. O. Gluzman, N. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications, Applied Mathematics Letters, 39 (2015), 19-21.  doi: 10.1016/j.aml.2014.08.006.  Google Scholar [13] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, Theory, Methods and Applications, 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar [14] N. V. Gorban, O. V. Kapustyan, P. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Continuous and Distributed Systems. 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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 [20] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar [21] A. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Sructure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, 211 (2014), 163-180.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar [22] 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 and Continuous Dynamical Systems, Series A, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar [23] 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 [24] P. O. Kasyanov, V. S. Mel'nik and S. Toscano, Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{λ_0}$-pseudomonotone maps, Journal of Differential Equations, 249 (2010), 1258-1287.  doi: 10.1016/j.jde.2010.05.008.  Google Scholar [25] P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar [26] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar [27] P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar [28] P. E. Kloeden, P. Marin-Rubio and J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3d navier-stokes and reaction-diffusion equations, Set-Valued and Variational Analysis, 21 (2013), 517-540.  doi: 10.1007/s11228-012-0228-x.  Google Scholar [29] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, (International Series of Monographs on Pure and Applied Mathematics, Vol. 45) Oxford/London/New York/Paris, 1964. Google Scholar [30] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Cambridge, 1991. Google Scholar [31] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and generalized differential equations, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar [32] V. S. Mel'nik and J. Valero, On global attractors of multivalued semiprocesses and non-autonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar [33] S. Migórski and A. Ochal, Optimal Control of Parabolic Hemivariational Inequalities, Journal of Global Optimization, 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar [34] S. Migórski, Boundary hemivariational inequalities of hyperbolic type and applications, Journal of Global Optimization, 31 (2005), 505-533.  doi: 10.1007/s10898-004-7021-9.  Google Scholar [35] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions, Birkhauser, Basel, 1985. Google Scholar [36] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dyn. Diff. Eq., 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar [37] J. Smoller, Shock Waves and Reaction-Diffusion Equations (Grundlehren der Mathematischen Wissenschaften) Springer-Verlag, New York, 1983. Google Scholar [38] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Verlag, Birkhäuser, 2001. Google Scholar [39] R. Temam, Navier-Stokes Equations North-Holland, Amsterdam, 1979. Google Scholar [40] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988. Google Scholar [41] J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J Math Anal Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar [42] M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor for a dissipative reaction-diffusion system, Doklady Mathematics, 82 (2010), 869-873.  doi: 10.1134/S1064562410060086.  Google Scholar [43] J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972. Google Scholar [44] M. Z. Zgurovsky, V. S. Mel'nik and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing II Springer, Berlin, 2011. Google Scholar [45] M. Z. Zgurovsky, P. O. Kasyanov and N. V. Zadoianchuk (Zadoyanchuk), Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Applied Mathematics Letters, 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.  Google Scholar [46] M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing III Springer, Berlin, 2012. Google Scholar [47] M. Z. Zgurovsky and P. O. Kasyanov, Multivalued dynamics of solutions for autonomous operator differential equations in strongest topologies, Continuous and Distributed Systems. Theory and Applications. Solid Mechanics and its Applications, 211 (2014), 149-162.  doi: 10.1007/978-3-319-03146-0_11.  Google Scholar [48] M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in Nonsmooth systems with applications for Earth data processing: Uniform trajectory attractors for non-autonomous evolution inclusions solutions with pointwise pseudomonotone mappings, Advances in Global Optimization, Springer Proceedings in Mathematics and Statistics, 95 (2015), 283-294.  doi: 10.1007/978-3-319-08377-3_28.  Google Scholar

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##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (Russian), Nauka, Moscow, 1989. Google Scholar [2] F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar [3] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the NavierStokes equations. Nonlinear Science, 7 (1997), 475{502 Erratum, ibid 8: 233,1998. Corrected version appears in Mechanics: From Theory to Computation. Springer Verlag, (2000), 447{ 474. Google Scholar [4] V. V. Chepyzhov and M. I. Vishik, Trajectory attractors for evolution equations, C.R.Acad.Sci. Paris. Serie I, 321 (1995), 1309-1314.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Evolution equations and their trajectory attractors, J. Math. Pures Appl., 76 (1997), 913-964.  doi: 10.1016/S0021-7824(97)89978-3.  Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Trajectory and global attractors for 3D Navier-Stokes system, Mat. Zametki., 71 (2002), 177-193.  doi: 10.1023/A:1014190629738.  Google Scholar [7] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence RI, 2002. Google Scholar [8] V. V. Chepyzhov and M. I. Vishik, Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time, Discrete and Continuous Dynamical Systems, 27 (2010), 1498-1509.  doi: 10.3934/dcds.2010.27.1493.  Google Scholar [9] F. H. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, Inc. , New York, 1983. Google Scholar [10] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications Kluwer Academic/Plenum Publishers, Boston, 2003. Google Scholar [11] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1975. Google Scholar [12] M. O. Gluzman, N. V. Gorban and P. O. Kasyanov, Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications, Applied Mathematics Letters, 39 (2015), 19-21.  doi: 10.1016/j.aml.2014.08.006.  Google Scholar [13] N. V. Gorban, O. V. Kapustyan and P. O. Kasyanov, Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, Theory, Methods and Applications, 98 (2014), 13-26.  doi: 10.1016/j.na.2013.12.004.  Google Scholar [14] N. V. Gorban, O. V. Kapustyan, P. O. Kasyanov and L. S. Paliichuk, On global attractors for autonomous damped wave equation with discontinuous nonlinearity, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2014), 221-237.  doi: 10.1007/978-3-319-03146-0_16.  Google Scholar [15] N. V. Gorban and P. O. Kasyanov, On regularity of all weak solutions and their attractors for reaction-diffusion inclusion in unbounded domain, Continuous and Distributed Systems. Theory and Applications, Solid Mechanics and its Applications, 211 (2013), 205-220.  doi: 10.1007/978-3-319-03146-0_15.  Google Scholar [16] J. K. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, RI, 1988. Google Scholar [17] G. Iovane, A. V. Kapustyan and J. Valero, Asymptotic behavior of reaction-diffusion equations with non-damped impulsive effects, Nonlinear Analysis, 68 (2008), 2516-2530.  doi: 10.1016/j.na.2007.02.002.  Google Scholar [18] A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, Journal of Differential Equations, 240 (2007), 249-278.  doi: 10.1016/j.jde.2007.06.008.  Google Scholar [19] A. 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 [20] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar [21] A. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Sructure of uniform global attractor for general non-autonomous reaction-diffusion system, Continuous and Distributed Systems: Theory and Applications, Solid Mechanics and Its Applications, 211 (2014), 163-180.  doi: 10.1007/978-3-319-03146-0_12.  Google Scholar [22] 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 and Continuous Dynamical Systems, Series A, 34 (2014), 4155-4182.  doi: 10.3934/dcds.2014.34.4155.  Google Scholar [23] 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 [24] P. O. Kasyanov, V. S. Mel'nik and S. Toscano, Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{λ_0}$-pseudomonotone maps, Journal of Differential Equations, 249 (2010), 1258-1287.  doi: 10.1016/j.jde.2010.05.008.  Google Scholar [25] P. O. Kasyanov, Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity, Cybernetics and Systems Analysis, 47 (2011), 800-811.  doi: 10.1007/s10559-011-9359-6.  Google Scholar [26] P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Mathematical Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar [27] P. O. Kasyanov, L. Toscano and N. V. Zadoianchuk, Regularity of weak solutions and their attractors for a parabolic feedback control problem, Set-Valued and Variational Analysis, 21 (2013), 271-282.  doi: 10.1007/s11228-013-0233-8.  Google Scholar [28] P. E. Kloeden, P. Marin-Rubio and J. Valero, The envelope attractor of non-strict multivalued dynamical systems with application to the 3d navier-stokes and reaction-diffusion equations, Set-Valued and Variational Analysis, 21 (2013), 517-540.  doi: 10.1007/s11228-012-0228-x.  Google Scholar [29] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, (International Series of Monographs on Pure and Applied Mathematics, Vol. 45) Oxford/London/New York/Paris, 1964. Google Scholar [30] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Cambridge, 1991. Google Scholar [31] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and generalized differential equations, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar [32] V. S. Mel'nik and J. Valero, On global attractors of multivalued semiprocesses and non-autonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.  Google Scholar [33] S. Migórski and A. Ochal, Optimal Control of Parabolic Hemivariational Inequalities, Journal of Global Optimization, 17 (2000), 285-300.  doi: 10.1023/A:1026555014562.  Google Scholar [34] S. Migórski, Boundary hemivariational inequalities of hyperbolic type and applications, Journal of Global Optimization, 31 (2005), 505-533.  doi: 10.1007/s10898-004-7021-9.  Google Scholar [35] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions, Birkhauser, Basel, 1985. Google Scholar [36] G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dyn. Diff. Eq., 8 (1996), 1-33.  doi: 10.1007/BF02218613.  Google Scholar [37] J. Smoller, Shock Waves and Reaction-Diffusion Equations (Grundlehren der Mathematischen Wissenschaften) Springer-Verlag, New York, 1983. Google Scholar [38] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Verlag, Birkhäuser, 2001. Google Scholar [39] R. Temam, Navier-Stokes Equations North-Holland, Amsterdam, 1979. Google Scholar [40] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988. Google Scholar [41] J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J Math Anal Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar [42] M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Strong trajectory attractor for a dissipative reaction-diffusion system, Doklady Mathematics, 82 (2010), 869-873.  doi: 10.1134/S1064562410060086.  Google Scholar [43] J. 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