April  2019, 24(4): 1961-1987. doi: 10.3934/dcdsb.2018257

Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China

* Corresponding author

Dedicated to the memory of professor V.S. Mel'nik

Received  May 2017 Revised  May 2018 Published  October 2018

Fund Project: This work was supported by NSF of China (Grants No. 41875084, 11571153), and the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2016-100 and lzujbky-2018- it58

We first prove the existence of a compact positively invariant set which exponentially attracts any bounded set for abstract multi-valued semidynamical systems. Then, we apply the abstract theory to handle retarded ordinary differential equations and lattice dynamical systems, as well as reactiondiffusion equations with infinite delays. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true.

Citation: Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
References:
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I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pur. Appl. Anal., 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.  Google Scholar

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A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons, New-York, 1994.  Google Scholar

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M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete Contin. Dynam. Systems., 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.  Google Scholar

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M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

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S. GattiM. GrasselliV. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete Contin. Dyn. Syst., 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.  Google Scholar

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M. Grasselli and D. Pražák, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.  Google Scholar

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M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483.  doi: 10.1007/s00028-005-0199-6.  Google Scholar

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S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.  Google Scholar

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P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

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A. V. Kapustyan and V. S. Mel'nik, Global attractors of multivalued semidynamical systems and their approximation, Cybernet. Syst. Anal., 34 (1998), 719-725.  doi: 10.1007/BF02667045.  Google Scholar

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A. V. Kapustyan and V. S. Mel'nik, On the global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.   Google Scholar

[24]

Y. J. LiH. Q. Wu and T. G. Zhao, Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications, Nonlinear Anal., 75 (2012), 6297-6305.  doi: 10.1016/j.na.2012.07.003.  Google Scholar

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V. S. Mel'nik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Values Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[26]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.  Google Scholar

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J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.  doi: 10.1016/j.jmaa.2010.04.074.  Google Scholar

[28]

Y. F. Shao and Y. H. Zhou, Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays, Nonlinear Anal., 74 (2011), 1107-1118.  doi: 10.1016/j.na.2010.09.042.  Google Scholar

[29]

M. Y. Sui and Y. J. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.  Google Scholar

[30]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.  Google Scholar

[31]

Y. J. Wang and M. Y. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptotic. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.  Google Scholar

[32]

M. Z. ZgurovskyPavlo O. Kasyanov and N. V. Zadoianchuk, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Appl. Math. Lett., 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.  Google Scholar

[33]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.  Google Scholar

[34]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.  Google Scholar

[35]

J. ZhangP. E. KloedenM. H. Yang and C. K. Zhong, Global exponential κ-dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148.  Google Scholar

[36]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.  Google Scholar

show all references

References:
[1]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[2]

A. V. Babin and B. Nicolaenko, Exponential attractors of reaction-diffusion systems in an unbounded domain, J. Dynam. Differential Equations, 7 (1995), 567-590.  doi: 10.1007/BF02218725.  Google Scholar

[3]

T. CaraballoP. Marí­n-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.  doi: 10.1023/A:1024422619616.  Google Scholar

[4]

T. CaraballoP. Marí­n-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations, 208 (2005), 9-41.  doi: 10.1016/j.jde.2003.09.008.  Google Scholar

[5]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[6]

T. CaraballoF. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar

[7]

C. Cavaterra and M. Grasselli, Robust exponential attractors for population dynamics models with infinite time delay, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1051-1076.  doi: 10.3934/dcdsb.2006.6.1051.  Google Scholar

[8]

D. N. Cheban and D. S. Fakeeh, Global attractors of infinite-dimensional dynamical systems â…¢, (Russian), Izv. Akad. Nauk Respub. Moldova Mat., (1995), 3-13,113,115.   Google Scholar

[9]

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

[10]

I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pur. Appl. Anal., 14 (2015), 1685-1704.  doi: 10.3934/cpaa.2015.14.1685.  Google Scholar

[11]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[12]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons, New-York, 1994.  Google Scholar

[13]

M. Efendiev and A. Miranville, Finite dimensional attractors for a class of reaction-diffusion equations in Rn with a strong nonlinearity, Discrete Contin. Dynam. Systems., 5 (1999), 399-424.  doi: 10.3934/dcds.1999.5.399.  Google Scholar

[14]

M. EfendievA. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.  doi: 10.1002/mana.200310186.  Google Scholar

[15]

S. GattiM. GrasselliV. Pata and M. Squassina, Robust exponential attractors for a family of nonconserved phase-field systems with memory, Discrete Contin. Dyn. Syst., 12 (2005), 1019-1029.  doi: 10.3934/dcds.2005.12.1019.  Google Scholar

[16]

M. Grasselli and D. Pražák, Exponential attractors for a class of reaction-diffusion problems with time delays, J. Evol. Equ., 7 (2007), 649-667.  doi: 10.1007/s00028-007-0326-7.  Google Scholar

[17]

M. Grasselli and V. Pata, Robust exponential attractors for a phase-field system with memory, J. Evol. Equ., 5 (2005), 465-483.  doi: 10.1007/s00028-005-0199-6.  Google Scholar

[18]

J. K. Hale, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, New York, 1977. Google Scholar

[19]

S. Habibi, Estimates on the dimension of an exponential attractor for a delay differential equation, Math. Slovaca, 64 (2014), 1237-1248.  doi: 10.2478/s12175-014-0272-0.  Google Scholar

[20]

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

[21]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[22]

A. V. Kapustyan and V. S. Mel'nik, Global attractors of multivalued semidynamical systems and their approximation, Cybernet. Syst. Anal., 34 (1998), 719-725.  doi: 10.1007/BF02667045.  Google Scholar

[23]

A. V. Kapustyan and V. S. Mel'nik, On the global attractors of multivalued semidynamical systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.   Google Scholar

[24]

Y. J. LiH. Q. Wu and T. G. Zhao, Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications, Nonlinear Anal., 75 (2012), 6297-6305.  doi: 10.1016/j.na.2012.07.003.  Google Scholar

[25]

V. S. Mel'nik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Values Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[26]

F. Morillas and J. Valero, Peano's theorem and attractors for lattice dynamical systems, Internat. J. Bifur. Chaos, 19 (2009), 557-578.  doi: 10.1142/S0218127409023196.  Google Scholar

[27]

J. C. Oliveira and J. M. Pereira, Global attractor for a class of nonlinear lattices, J. Math. Anal. Appl., 370 (2010), 726-739.  doi: 10.1016/j.jmaa.2010.04.074.  Google Scholar

[28]

Y. F. Shao and Y. H. Zhou, Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays, Nonlinear Anal., 74 (2011), 1107-1118.  doi: 10.1016/j.na.2010.09.042.  Google Scholar

[29]

M. Y. Sui and Y. J. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.  Google Scholar

[30]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differential Equations, 232 (2007), 573-622.  doi: 10.1016/j.jde.2006.07.005.  Google Scholar

[31]

Y. J. Wang and M. Y. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptotic. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.  Google Scholar

[32]

M. Z. ZgurovskyPavlo O. Kasyanov and N. V. Zadoianchuk, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Appl. Math. Lett., 25 (2012), 1569-1574.  doi: 10.1016/j.aml.2012.01.016.  Google Scholar

[33]

S. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's A-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.  Google Scholar

[34]

S. F. Zhou and X. Y. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dyn. Diff. Equat., 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.  Google Scholar

[35]

J. ZhangP. E. KloedenM. H. Yang and C. K. Zhong, Global exponential κ-dissipative semigroups and exponential attraction, Discrete Contin. Dyn. Syst., 37 (2017), 3487-3502.  doi: 10.3934/dcds.2017148.  Google Scholar

[36]

C. K. Zhong and W. S. Niu, On the Z2-index of the global attractor for a class of p-Laplacian equations, Nonlinear Anal., 73 (2010), 3698-3704.  doi: 10.1016/j.na.2010.07.022.  Google Scholar

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