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:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (Russian), Nauka, Moscow, 1989.

[2]

F. BalibreaT. CaraballoP. 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.

[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.

[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.

[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.

[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.

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence RI, 2002.

[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.

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, Inc. , New York, 1983.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications Kluwer Academic/Plenum Publishers, Boston, 2003.

[11]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1975.

[12]

M. O. GluzmanN. 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.

[13]

N. V. GorbanO. 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.

[14]

N. V. GorbanO. V. KapustyanP. 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.

[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.

[16]

J. K. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, RI, 1988.

[17]

G. IovaneA. 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.

[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.

[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.

[20]

O. V. KapustyanP. 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.

[21]

A. V. KapustyanP. O. KasyanovJ. 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.

[22]

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 and Continuous Dynamical Systems, Series A, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[23]

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.

[24]

P. O. KasyanovV. 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.

[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.

[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.

[27]

P. O. KasyanovL. 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.

[28]

P. E. KloedenP. 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.

[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.

[30]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Cambridge, 1991.

[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.

[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.

[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.

[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.

[35]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions, Birkhauser, Basel, 1985.

[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.

[37]

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Grundlehren der Mathematischen Wissenschaften) Springer-Verlag, New York, 1983.

[38]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Verlag, Birkhäuser, 2001.

[39]

R. Temam, Navier-Stokes Equations North-Holland, Amsterdam, 1979.

[40]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988.

[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.

[42]

M. I. VishikS. 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.

[43]

J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972.

[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.

[45]

M. Z. ZgurovskyP. 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.

[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.

[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.

[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.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (Russian), Nauka, Moscow, 1989.

[2]

F. BalibreaT. CaraballoP. 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.

[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.

[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.

[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.

[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.

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics American Mathematical Society, Providence RI, 2002.

[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.

[9]

F. H. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, Inc. , New York, 1983.

[10]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications Kluwer Academic/Plenum Publishers, Boston, 2003.

[11]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen Akademie-Verlag, Berlin, 1975.

[12]

M. O. GluzmanN. 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.

[13]

N. V. GorbanO. 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.

[14]

N. V. GorbanO. V. KapustyanP. 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.

[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.

[16]

J. K. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, RI, 1988.

[17]

G. IovaneA. 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.

[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.

[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.

[20]

O. V. KapustyanP. 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.

[21]

A. V. KapustyanP. O. KasyanovJ. 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.

[22]

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 and Continuous Dynamical Systems, Series A, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155.

[23]

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.

[24]

P. O. KasyanovV. 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.

[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.

[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.

[27]

P. O. KasyanovL. 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.

[28]

P. E. KloedenP. 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.

[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.

[30]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations Cambridge University Press, Cambridge, 1991.

[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.

[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.

[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.

[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.

[35]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions, Birkhauser, Basel, 1985.

[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.

[37]

J. Smoller, Shock Waves and Reaction-Diffusion Equations (Grundlehren der Mathematischen Wissenschaften) Springer-Verlag, New York, 1983.

[38]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Verlag, Birkhäuser, 2001.

[39]

R. Temam, Navier-Stokes Equations North-Holland, Amsterdam, 1979.

[40]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Appl. Math. Sci. , Springer-Verlag, New York, 1988.

[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.

[42]

M. I. VishikS. 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.

[43]

J. Warga, Optimal Control of Differential and Functional Equations Academic Press, 1972.

[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.

[45]

M. Z. ZgurovskyP. 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.

[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.

[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.

[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.

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